Quasilinear parabolic and elliptic systems with mixed quasimonotone functions

J. Differential Equations 255 (2013) 1515–1553

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Journal of Differential Equations www.elsevier.com/locate/jde

Quasilinear parabolic and elliptic systems with mixed quasimonotone functions C.V. Pao a , W.H. Ruan b,∗ a b

Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, United States Department of Mathematics, Computer Science and Statistics, Purdue University Calumet, Hammond, IN 46323-2094, United States

a r t i c l e

i n f o

Article history: Received 29 June 2012 Available online 3 June 2013 MSC: primary 35K50, 35J55 secondary 35K65, 35K57 Keywords: Degenerate parabolic and elliptic system Global existence Maximal–minimal solutions Global attractor Predator–prey problems

a b s t r a c t This paper deals with a class of quasilinear parabolic and elliptic systems with mixed quasimonotone reaction functions. The boundary condition in the system may be Dirichlet, nonlinear, or a combination of these two types. The elliptic operators in the system are allowed to be degenerate. The aim is to show the existence and uniqueness of a classical solution to the parabolic system, the existence of maximal and minimal solutions or quasisolutions of the elliptic system, and the asymptotic behavior of the solution of the parabolic system. This consideration leads to a global attractor of the parabolic system as well as an one-sided stability of the maximal and minimal solutions. Applications of these results are given to three models arising from biology and ecology where diffusion coefficients are density-dependent and are degenerate. These applications exhibit quite distinct dynamical behavior of the population species between degenerate density-dependent diffusion and constant diffusion. © 2013 Elsevier Inc. All rights reserved.

1. Introduction Coupled system of parabolic and elliptic equations arises from many fields of applied sciences and has been given extensive attention in both theory and application. Most discussions in the earlier literature are for systems of semilinear equations under either Dirichlet boundary condition or Neumann–Robin boundary condition. In recent years, attention has been given to quasilinear parabolic and elliptic equations and various methods have been proposed to treat many different aspects of the problem. Using the method of upper and lower solutions in two recent articles [28,30] the authors investigated a class of quasilinear parabolic and elliptic systems under either Dirichlet boundary

*

Corresponding author. E-mail address: [email protected] (W.H. Ruan).

0022-0396/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2013.05.015

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condition or nonlinear boundary condition. The main concerns in these works are the existence of a global solution to the quasilinear parabolic system and the asymptotic behavior of the solution in relation to the maximal and minimal solutions of the corresponding quasilinear elliptic system. A major requirement in the above papers is that the “reaction functions” in the equation or in the boundary condition possess a quasimonotone nondecreasing property between a pair of upper and lower solutions. This requirement excludes its application to many other type of reaction–diffusion problems where the diffusion coefficients are density-dependent and the reaction functions are not quasimonotone nondecreasing. The purpose of this paper is to extend the work in [28,30] to a class of mixed quasimonotone reaction functions and with mixed type of boundary conditions, including a system of parabolic-ordinary equations which arise often in more concrete problems. Of special concern is the degenerate case where some or all of the diffusion coefficients may vanish on the boundary of the domain. This extension makes it possible to apply the results to a much larger class of reaction– diffusion systems where the reaction function is mixed quasimonotone and the diffusion coefficients are density-dependent and possibly degenerate. The parabolic system under consideration is given in the form

    ∂ u i /∂ t − ∇ · a∗i D i (u i )∇ u i + b∗i · D i (u i )∇ u i = f i (t , x, u),

i = 1, 2, . . . , N

u i (t , x) = h i (t , x),

i = 1, 2, . . . , n 0 − 1

D i (u i )∂ u i /∂ ν = g i (t , x, u),

i = n0 , . . . , N

u i (0, x) = ψi (x),

i = 1, . . . , N

(t > 0, x ∈ Ω), (t > 0, x ∈ ∂Ω), (x ∈ Ω),

(1.1)

where u = (u 1 , . . . , u N ), Ω is a bounded domain in Rn with boundary ∂Ω , ∂/∂ ν denotes the outward normal derivative on ∂Ω , and for each i = 1, 2, . . . , N, a∗i = a∗i (t , x), b∗i = b∗i (t , x), D i (u i ), f i (t , x, u), g i (t , x, u), h i (t , x) and ψi (x) are given functions satisfying the conditions in Hypothesis (H1 ) in Section 2. It is assumed that Ω is of class C 1+α for some α ∈ (0, 1). However, if n0 = N + 1 then Ω is only required to satisfy the outside strong sphere property (cf. [25, p. 16] or [30]). Notice that the boundary condition in (1.1) is of Dirichlet type for every i if n0 = N + 1, and it is nonlinear for every i if n0 = 1. The consideration of a general n0 gives more flexibility in the application to more specific problems. Moreover, we allow D i (0) = 0 for some or all i. This leads to a degenerate system at u i = 0, especially when the boundary condition is given by u i (t , x) = 0 on ∂Ω . To investigate the asymptotic behavior of the solution of (1.1), we also consider the corresponding quasilinear elliptic system

    −∇ · a∗i D i (u i )∇ u i + b∗i · D i (u i )∇ u i = f i (x, u), i = 1, 2, . . . , N u i (x) = h i (x),

i = 1, 2, . . . , n 0 − 1

D i (u i )∂ u i /∂ ν = g i (x, u),

i = n0 , . . . , N

(x ∈ Ω), (x ∈ ∂Ω).

(1.2)

It is obvious that the above system is degenerate if D i (0) = 0 and h i (x) = 0 for some or all i = 1, . . . , n0 − 1. In many concrete reaction–diffusion problems some of the density functions are time and space dependent and others are time-dependent but are spatially homogeneous. This leads to a coupled system of parabolic-ordinary system in the form

    ∂ u i /∂ t − ∇ · a∗i D i (u i )∇ u i + b∗i · D i (u i )∇ u i = f i (t , x, u), i = 1, . . . , n1 ∂ u i /∂ t = f i (t , x, u),

i = n 1 + 1, . . . , N

u i (t , x) = h i (t , x),

i = 1, . . . , n 0 − 1

D i (u i )∂ u i /∂ ν = g i (t , x, u),

i = n0 , . . . , n1

u i (0, x) = ψi (x),

i = 1, . . . , N

(t > 0, x ∈ Ω), (t > 0, x ∈ ∂Ω), (x ∈ Ω).

(1.3)

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The above system may be considered as a special case of (1.1) with D i (u i ) = 0 and without the boundary condition for i = n1 + 1, . . . , N. As in (1.1) we also consider the corresponding elliptic system which is given by

    −∇ · a∗i D i (u i )∇ u i + b∗i · D i (u i )∇ u i = f i (x, u), i = 1, . . . , n1 f i (x, u) = 0,

i = n 1 + 1, . . . , N

u i (x) = h i (x),

i = 1, . . . , n 0 − 1

D i (u i )∂ u i /∂ ν = g i (x, u),

i = n0 , . . . , n1

(x ∈ Ω), (x ∈ ∂Ω),

(1.4)

where a∗i = a∗i (x), b∗i = b∗i (x) are independent of t. The purpose of this paper is to investigate: (i) the existence and uniqueness of a classical global solution to (1.1) and (1.3), (ii) the existence of quasisolutions or maximal and minimal solutions of (1.2) and (1.4), and (iii) the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions or quasisolutions as t → ∞. Of particular concern is the degenerate case where D i (0) = 0 and the boundary condition is u i (x) = 0 on ∂Ω . Application of these results is given to three model problems arising from biology and ecology where the diffusion coefficients are degenerate and the reaction functions are mixed quasimonotone. This application gives some quite interesting distinct dynamical property of the system when compared with the corresponding semilinear system with constant diffusion coefficient. Quasilinear parabolic equations in the form of (1.1) have been investigated by many researchers, and most of the discussions in the earlier literature are for scalar equations with N = 1 (cf. [1,12, 18,23,37] and references therein). In recent years the main concerns are for the existence and regularity of a global solution, [2–4,7,10–13,22,24,34–36,38,40,42,48], large time behavior of the solution (cf. [1,5,24,34,36]) and finite-time blow up of the solution (cf. [10,14,35,42]). This kind of problem has been extended in [9,14–16,19,20,29,37,41,43,47] for coupled system of two or more reaction–diffusion equations where the diffusion coefficients are of porous medium type and the reaction functions are of special kind of quasimonotone nondecreasing functions. Also treated in the literature are for the existence and multiplicity of positive solutions to the corresponding quasilinear elliptic system (cf. [21,26,28,30]). The work in [27,39] gives some numerical treatment of the scalar equations by the method of upper and lower solutions. More recently the authors investigated a general coupled system of N equations in the form of (1.1) with either Dirichlet boundary condition or nonlinear boundary condition where the reaction functions f(t , x, u) and g(t , x, u) are quasimonotone nondecreasing (cf. [28,30]). Much of the discussions in the above papers are devoted to degenerate parabolic equations with positive initial functions in Ω . It is well-known that the solution of degenerate parabolic equation may exhibit quite different properties from the solution of nondegenerate equation, such as finite propagation speed, non-smoothness property, and distinct asymptotic behavior of the solution (cf. [1,6,12, 18,23,28,30,37]). In particular, degenerate parabolic equation may fail to possess a classical solution, and therefore various forms of weak solutions are introduced, including the concept of viscosity solution (cf. [19,31–33,37]). The nonexistence of a classical solution can also occur if the initial function vanishes in a part of the spatial domain. Consider, for example, the scalar porous medium problem

∂ u /∂ t − ∇ 2 u σ = 0 in Q T , where

u (t , x, y ) = 0 on S T ,

σ > 0 is a constant, ψ(x, y ) =

1 [r 2 − (x2 + 16 0

u (0, x, y ) = ψ(x, y )

in Ω,

(1.5)

y 2 )]+ for some r0 > 0, and Ω is a bounded domain

in R2 containing the disk region Ω0 ≡ {(x, y ): x2 + y 2  r12 }, where r1 > r0 is a constant, [s]+ = s if s  0 and [s]+ = 0 if s < 0. It is easy to verity that the (local)solution of (1.5) for the case σ = 2 is given by



−1 

u (t , x, y ) = 16(t + 1)





r02 (t + 1)1/2 − x2 + y 2 +

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for t < T where T < (r1 /r0 )4 − 1. Clearly, this solution is continuous in Q T but it is not smooth there. The same conclusion holds true if the Dirichlet condition u = 0 in (1.5) is replaced by the Neumann condition ∂ u /∂ ν = 0 on S T . The non-smoothness property of the solution in the above example demonstrates that in order to ensure the existence of a classical solution to (1.1) where D i (0) = 0 for some i it is necessary, in general, to require that the corresponding initial function ψi to be positive in Ω (but not necessarily on ∂Ω ). This condition has been assumed in [28,30] and will be assumed for the present problem (1.1) (see Hypothesis (H1 )(iii) in Section 2). The plan of the paper is as follows: In Section 2, we show the existence and uniqueness of a classical solution to (1.1) (and (1.3)) using the method of upper and lower solutions. Section 3 is devoted to the existence of a pair of quasisolutions to (1.3) (and (1.4)) and the existence of a maximal and a minimal solution for quasimonotone nondecreasing reaction functions. In Section 4, we show the one-sided stability property of the maximal and minimal solutions for quasimonotone nondecreasing functions. In the general case of mixed quasimonotone functions we show that the sector between the quasisolutions is a global attractor of the time-dependent system. Applications are given in Section 5 to three model problems arising from population growth problems which demonstrate some distinct dynamical behavior between degenerate diffusion and constant diffusion. 2. Quasilinear parabolic system Let Q T = (0, T ] × Ω , Q T = [0, T ] × Ω , S T = (0, T ] × ∂Ω , and let C m ( Q ), C α ( Q ) be the respective function spaces of m-times continuously differentiable and Hölder continuous functions in Q , where T > 0 is an arbitrary constant and Q represents a domain or a sector between two functions. For vector functions of N-components we denote the above function spaces by C m ( Q ) and C α ( Q ), respectively. Similar function spaces will be used throughout the paper. To show the existence of a classical solution to (1.1) we use the method of upper and lower solutions for mixed quasimonotone functions. Recall that a vector function





f(·, u) ≡ f 1 (·u), . . . , f N (·, u)

is said to have a mixed quasimonotone property in a subset S of R N if for each i = 1, . . . , N there exist nonnegative integers ai , b i with ai + b i = N − 1 such that f i (·, u i , [u]ai , [u]bi ) is nondecreasing in [u]ai and is nonincreasing in [u]bi for all u ∈ S , where u is written in the split form u = (u i , [u]ai , [u]bi ), and [u]ai and [u]bi are the vectors with ai and b i components of u, respectively (cf. [25]). In particular, if b i = 0 for all i then f(·, u) is said to be quasimonotone nondecreasing in S . In this situation, f(·, u) possesses the property

∂ fi (u 1 , . . . , u N )  0 for u ∈ S and all j = i , i = 1, . . . , N . ∂u j For the general mixed quasimonotone functions f(t , x, u) and g(t , x, u) we write Problem (1.1) in the form

    ∂ u i /∂ t − ∇ · a∗i D i (u i )∇ u i + b∗i · D i (u i )∇ u i i = 1, . . . , N   = f i t , x, u i , [u]ai , [u]bi , u i (t , x) = h i (t , x),







 (t , x) ∈ Q T ,

i = 1, . . . , n 0 − 1 

D i (u i )∂ u i /∂ ν = g i t , x, u i , [u]c i , [u]di ,

i = n0 , . . . , N

u i (0, x) = ψi (x),

i = 1, . . . , N

where ai , b i , c i and di are nonnegative integers satisfying

ai + b i = c i + di = N − 1 for all i = 1, . . . , N .

 (t , x) ∈ S T ,

(x ∈ Ω),

(2.1)

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

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The definition of upper and lower solutions depends on the mixed quasimonotone properties of f(t , x, u) and g(t , x, u) and is given by the following:

˜ ≡ (u˜ 1 , . . . , u˜ N ), uˆ ≡ (uˆ 1 , . . . , uˆ N ) in C ( Q T ) ∩ C 2 ( Q T ) are called Definition 2.1. A pair of functions u ˜  uˆ > 0 in Q T and if coupled upper and lower solutions of (2.1) if u

    ∂ u˜ i /∂ t − ∇ · a∗i D i (u˜ i )∇ u˜ i + b∗i · D i (u˜ i )∇ u˜ i    f i t , x, u˜ i , [u˜ ]ai , [uˆ ]bi ,   i = 1, . . . , N   ∂ uˆ i /∂ t − ∇ · a∗i D i (uˆ i )∇ uˆ i + b∗i · D i (uˆ i )∇ uˆ i    f i t , x, uˆ i , [uˆ ]ai , [u˜ ]bi , u˜ i (t , x)  h i (t , x)  uˆ i (t , x),









i = 1, . . . , n 0 − 1

˜ ]ci , [uˆ ]di , D i (u˜ i )∂ u˜ i /∂ ν  g i t , x, u˜ i , [u

i = n0 , . . . , N

ˆ ]ci , [u˜ ]di , D i (uˆ i )∂ uˆ i /∂ ν  g i t , x, uˆ i , [u

i = n0 , . . . , N

u˜ i (0, x)  ψi (x)  uˆ i (0, x),

i = 1, . . . , N



 (t , x) ∈ Q T ,



 (t , x) ∈ S T ,

(x ∈ Ω).

(2.2)

˜ and uˆ are, in general, coupled and a solution It is clear from the above definition that the pair u of (2.1) is neither an upper solution nor a lower solution. However, if b i = di = 0 for all i, that is, if f(t , x, u) and g(t , x, u) are both quasimonotone nondecreasing, then upper and lower solutions are not coupled and are called ordered upper and lower solutions. In this situation, a solution of (2.1) is an upper solution as well as a lower solution. It is to be noted that in the above definition it is ˆ > 0 in Q T (not necessarily on S T ). This implies that a solution between upper and required that u lower solutions is always positive in Q T . However, it may vanish on S T even if D i (0) = 0 for some or ˜ uˆ we set every i. For a given pair of coupled or ordered upper and lower solutions u,



S i ≡ u i ∈ C α ( Q T ): uˆ i  u i  u˜ i

 S ≡ u ∈ C α ( Q T ): uˆ  u  u˜ . 



(i = 1, . . . , N ), (2.3)

To ensure the existence of a classical solution to (2.1) we assume that there exist a pair of coupled ˜ uˆ and impose the following hypothesis: upper and lower solutions u,

(H1 )

(i) For each i, a∗i (t , x) ∈ C α /2,1 ( Q T ), (b∗i )(l) (t , x), l = 1, . . . , n (the components of b∗i ) and f i (t ,

x, ·) are in C α /2,α ( Q T ), a∗i (t , x)  a0 > 0, h i (t , x) ∈ C α /2,α ( S T ), g i (t , x, ·) ∈ C (1+α )/2,1+α ( S T ), and f i (·, u) and g i (·u) are in C (1) (S ). (ii) D i (u i ) ∈ C 1+α ( S i ), D i (u i ) > 0 for 0 < u i  u˜ i , D i (0)  0 for i = 1, . . . , N, and D i (u i )  0 for u i near zero. Moreover, for i = n0 , . . . , N, either D i (0) > 0 or D i (0) = 0 and uˆ i  δi∗ for some constant δi∗ > 0. (iii) f(·, u) and g(·, u) are mixed quasimonotone in S , f i (t , x, 0)  0 for i = 1, . . . , n0 − 1, and (1)

there exist nonnegative functions c i

(1)

(2)

≡ c i (t , x), c i

(2)

≡ c i (t , x) such that

(1 )

c i D i (u i ) + (∂ f i /∂ u i )(t , x, u)  0 for u ∈ S (i = 1, . . . , N ), (2 )

c i D i (u i ) + (∂ g i /∂ u i )(t , x, u)  0 for u ∈ S (i = n0 , . . . , N ).

(2.4)

(iv) ψi ∈ C 2+α (Ω) ∩ C (Ω), ψi > 0 in Ω for i = 1, 2, . . . , N, and ψi (x) = h i (0, x) on ∂Ω for i = 1, 2, . . . , n0 − 1. In the above hypothesis we allow the possibility that D i (0) = 0 and h i (t , x) = 0 for some or all (t , x) ∈ S T . This implies that Problem (2.1) is degenerate on S T when D i (0) = 0 and hi (t , x) = 0 for

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some or all i = 1, . . . , n0 − 1. In fact, our main concern in the following discussion is the above degenerate case which is crucial in the study for many concrete degenerate parabolic and elliptic systems with homogeneous Dirichlet boundary conditions, including various porous medium type of reaction–diffusion systems. It is to be noted that condition (2.4) in Hypothesis (H1 )(iii) is trivially satisfied if D i (0) > 0. In the degenerate case it is also satisfied if (∂ f i /∂ u i )  0 and (∂ g i /∂ u i )  0 for u ∈ S. To develop a monotone iterative scheme for the existence of a solution to (2.1) we transform the problem into an equivalent system by defining

u i w i = I i [u i ] ≡

D i (s) ds

for u i ∈ S i (i = 1, . . . , N ).

(2.5)

0

Since dw i /du i = D i (u i ) > 0 for u i > 0 the inverse of w i , denoted by u i = qi ( w i ), exists and is an increasing function of w i . Moreover, from the relation

∂ u i /∂ t = q i ( w i )∂ w i /∂ t ,

∇ w i = D i (u i )∇ u i ,

∂ w i /∂ ν = D i (u i )∂ u i /∂ ν

we may write (2.1) in the equivalent form









q i ( w i )∂ w i /∂ t − ∇ · a∗i ∇ w i + b∗i · ∇ w i = f i t , x, u i , [u]ai , [u]bi ,

i = 1, . . . , N ,

w i (t , x) = h∗i (t , x),

i = 1, . . . , n 0 − 1,





∂ w i /∂ ν = g i t , x, u i , [u]ci , [u]di ,

i = n0 , . . . , N ,

w i (0, x) = ηi (x),

i = 1, . . . , N ,

u i = qi ( w i ),

i = 1, . . . , N ,

where q i ( w i ) = dqi /dw i = ( D i (u i ))−1 , h∗i = I i [h i ] and





w = I [u] = I 1 [u 1 ], . . . , I N [u N ] ,

(2.6)

ηi = I i [ψi ]. Let 



u = q(w) = q1 ( w 1 ), . . . q N ( w N ) .

˜,w ˜ ) = (u, I [u˜ ]) and (uˆ , w ˆ ) = (uˆ , I [uˆ ]) are coupled upper and lower It is easy to verify that the pair (u ˜,w ˜ )  (uˆ , w ˆ ) > (0, 0) in Q T and their components (u˜ i , w ˜ i ), solutions of (2.6) in the sense that (u ˆ i ) satisfy the inequalities (uˆ i , w

















˜ i )∂ w ˜ i /∂ t − ∇ · a∗i ∇ w ˜ i + b∗i · ∇ w ˜ i  f i t , x, u˜ i , [u˜ ]ai , [uˆ ]bi , q i ( w ˆ i )∂ w ˆ i /∂ t − ∇ · a∗i ∇ w ˆ i + b∗i · ∇ w ˆ i  f i t , x, uˆ i , [uˆ ]ai , [u˜ ]bi , q i ( w ˆ i (t , x), ˜ i (t , x)  h∗i (t , x)  w w 



i = 1, . . . , N , i = 1, . . . , n 0 − 1,

˜ i /∂ ν  g i t , x, u˜ i , [u˜ ]ci , [uˆ ]di , ∂w   ˆ i /∂ ν  g i t , x, uˆ i , [uˆ ]ci , [u˜ ]di , ∂w

i = n0 , . . . , N ,

ˆ i (0, x), ˜ i (0, x)  ηi (x)  w w

i = 1, . . . , N .

Define

  ˆ )  (u, w)  (u˜ , w ˜) , S × S = (u, w) ∈ C α ( Q T ) × C α ( Q T ); (uˆ , w   (1 ) L i w i = ∇ · a∗i ∇ w i − b∗i · ∇ w i − c i w i ,

(2.7)

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553





(1 )









(2 )





1521

F i t , x, u i , [u]ai , [u]bi = c i I i [u i ] + f i t , x, u i , [u]ai , [u]bi , G i t , x, u i , [u]c i , [u]di = c i I i [u i ] + g i t , x, u i , [u]c i , [u]di (1)

where c i

(2)

and c i

(2.8)

are the functions in (2.4). Since dI i /du i = D i (u i ) and by (2.4)

(1 )

∂ F i /∂ u i = c i D i (u i ) + (∂ f i /∂ u i )(·, u)  0,

(2 )

∂ G i /∂ u i = c i D i (u i ) + (∂ g i /∂ u i )(·, u)  0,

the mixed quasimonotone property of f(·, u) and g(·, u) implies that

















F i ·, u i , [u]ai , [v]bi  F i ·, v i , [v]ai , [u]bi ,

˜  u  v  uˆ . whenever u

G i ·, u i , [u]c i , [v]di  G i ·, v i , [v]c i , [u]di ,

(2.9)

Moreover, Problem (2.6) may be written as





q i ( w i )∂ w i /∂ t − L i w i = F i t , x, u i , [u]ai , [u]bi ,

i = 1, . . . , N ,

w i (t , x) = h∗i (t , x),

i = 1, . . . , n 0 − 1,

(2 )





∂ w i /∂ ν + c i w i = G i t , x, u i , [u]ci , [u]di ,

i = n0 , . . . , N ,

w i (0, x) = ηi (x)

i = 1, . . . , N ,

in Ω,

u i = qi ( w i ),

i = 1, . . . , N .

(2.10)

Although the function u in F i (·, u) (or G i (·, u)) can be replaced by q(w) we prefer to write it in the present form because F i (·, q(w)) may not be a C 1 -function in w i if D i (0) = 0. Notice that q i ( w i ) = ( D i (u i ))−1 so that q i ( w i ) is singular at w i = 0 if D i (0) = 0. ¯ (0) , w ¯ (0) ) = (u˜ , w ˜ ) and (u(0) , w(0) ) = (uˆ , w ˆ ) as a pair of coupled initial iterations we can Using (u ¯ (m) , w ¯ (m) }, {u(m) , w(m) } from the (nonlinear) iteration process construct two sequences {u



(m) 

 (m−1)  (m−1)   (m−1)   , = F i ·, u¯ i , u¯ , u ai bi i = 1, . . . , N ,         (m) ∂ w (i m) /∂ t − L i w (i m) = F i ·, u (i m−1) , u(m−1) a , u¯ (m−1) b , q i w i ¯i q i w

(m)

¯i ∂w

(m)

¯i /∂ t − L i w

i

i

(t , x) = h∗i (t , x),       ¯ (i m) = G i ·, u¯ (i m−1) , u¯ (m−1) c , u(m−1) d , ¯ (i m) /∂ ν + c i(2) w ∂w i i  (m−1)  (m−1)   (m−1)   (m) (2) (m) ¯ , ∂ w i /∂ ν + c i w i = G i ·, u i , u , u c d (m)

¯i w

(m)

(t , x) = w i

i

i = 1, . . . , n 0 − 1, i = n0 , . . . , N ,

i

(m)

(0, x) = w (i m) (0, x) = ηi (x),  (m)   (m)  (m) (m) ¯i , , u¯ i = qi w u i = qi w i ¯i w

i = 1, . . . , N , i = 1, . . . , N .

(2.11)

To show the existence and monotone property of these sequences we use some results from [28,30] for the scalar quasilinear parabolic problem

q i ( W i )∂ W i /∂ t − L i W i = P i (t , x),

αi ∂ W i /∂ ν + βi W i = H i (t , x), W i (0, x) = ηi (x),

(2.12)

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C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

where i is fixed, P i and H i have the same property as f i (t , x, ·) and g i (t , x, ·) in (H1 )(i) and either αi = 0, βi = 1 and H i (t , x)  0 (Dirichlet condition) or αi = 1, βi ≡ βi (t , x)  0 (Neumann–Robin condition). For the above scalar boundary problem we have the following results:

˜ i, W ˆ i be a pair of ordered upper and lower solutions of (2.12) with W ˆ i > 0 in Q T . Then Lemma 2.1. Let W ˜ i if αi = 0, βi = 1 and H i  0. The same is ˆ i  W∗  W Problem (2.12) has a unique solution W i∗ such that W i ˆ i  δ ∗ for some constant δ ∗ > 0. true for the case αi = 1, βi  0 if either D i (0) > 0 or W i i

Proof. The conclusion of the lemma for the case αi = 0, βi = 1 follows from Theorem 2.1 of [30] while that for the case αi = 1, βi  0 is a consequence of Theorem 2.1 of [28]. 2 We next show the existence and monotone property of the sequences governed by (2.11).

¯ (m) , w ¯ (m) }, {u(m) , w(m) } governed by (2.11) are well-defined and possess the Lemma 2.2. The sequences {u monotone property

        ˆ )  u(m) , w(m)  u(m+1) , w(m+1)  u¯ (m+1) , w ¯ (m+1)  u¯ (m) , w ¯ (m)  (u˜ , w ˜) (uˆ , w

(2.13)

for m = 1, 2, . . . .

˜ ) and (uˆ , w ˆ ), and define ˜ i ), (uˆ i , w ˆ i ) be the components of (u˜ , w Proof. Let (u˜ i , w

      (t , x) = F i t , x, u¯ (i m) (t , x), u¯ (m) (t , x) a , u(m) (t , x) b i i     (m)   (i = 1, . . . , N ), (m) (m) ¯ i (t , x) b P i (t , x) = F i t , x, u i (t , x), u(m) (t , x) a , u P¯ i

(m)

i

¯ H i

i

(t , x) = H i (t , x) = h∗i (t , x)       ¯ (m) (t , x) = G i t , x, u¯ (m) (t , x), u¯ (m) , u(m) H i i ci di   (m)   (m)   (m) (m) H i (t , x) = G i t , x, u i (t , x), u , u¯ c d (m)

(m)

i

(i = 1, . . . , n0 − 1), (i = n0 , . . . , N ).

i

(2.14)

It is clear from (2.9) that

P¯ i

(m)

(m) (m) (m) (t , x)  P i (t , x) and H¯ i (t , x)  H i (t , x) for i = 1, . . . , N when u¯ (m)  u(m) .

¯ (t , x). Since by (2.7), (2.8) and (2.14), Consider Problem (2.12) with P i (t , x) = P¯ i (t , x), H i (t , x) = H i ˜ i and w ˆ i satisfy the inequalities w (0)



(0)

   ( 0) , u(0) b = P¯ i (t , x), i       ˆ i  F i t , x, u (i 0) , u(0) a , u¯ (0) b = P (i 0) (t , x)  P¯ i(0) (t , x), ˆ i )∂ w ˆ i /∂ t − L i w q i ( w 

˜ i  F i t , x, u¯ i , u¯ (0) ˜ i )∂ w ˜ i /∂ t − L i w q i ( w ( 0)



ai i

i

˜ i (t , x)  h∗i (t , x)  w ˆ i (t , x), w       ˜ i  G i t , x, u¯ (i 0) , u¯ (0) c , u(0) d = H¯ i(0) (t , x), ˜ i /∂ ν + c i(2) w ∂w i i  (2 ) ( 0)  ( 0)   ( 0)   ( 0) ( 0) ˆ i /∂ ν + c i w ˆ i  G i t , x, u i , u c , u¯ ∂w = H i (t , x)  H¯ i (t , x), d i

i

˜ i (0, x)  ηi (x)  w ˆ i (0, x) w ˜ i and w ˆ i are ordered upper and lower solutions of (2.12) for the case P i (t , x) = we see that w (0) ¯ (0) (t , x). By Lemma 2.1, this problem has a unique solution, denoted by w ¯ (i 1) , P¯ i (t , x), H i (t , x) = H i

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

1523

(1)

˜ i . Similarly by considering Problem (2.12) with P i (t , x) = P (i 0) (t , x) and w (0) ˜ i, w ˆ i are also ordered upper and lower solutions. This implies the H i (t , x) = H i (t , x), the pair w (1) ˜ i . We next show that w ˆ i  w (i 1)  w ¯ (i 1)  w (i 1) for existence of a unique solution w i such that w (0) ( 0 ) ( 0 ) ( 0 ) ( 1 ) every i = 1, . . . , N. It is easily seen from P  P¯ and H  H¯ that w is a lower solution ¯i ˆi  w such that w

i

i

i

i

i

(0) ¯ (0) . Since w ¯ (i 1) may be considered as an upper solution of the of (2.12) when P i = P¯ i and H i = H i (1)

(1)

¯i same problem we conclude from Lemma 2.1 of [30] that w (1)

show this result for i = n0 , . . . , N we let zi



(1)

¯i =w

 wi

(1)

for the case i = 1, . . . , n0 − 1. To

− w i . Then

(1 ) 

  ¯ (i 1) /∂ t − q i w (i 1) ∂ w (i 1) /∂ t − L i z(i 1) = P¯ i (t , x) − P i (t , x)  0, ∂w

¯i q i w (1 )

(2 ) (1 )

∂ zi /∂ ν + c i zi

= H¯ i (t , x) − H i (t , x)  0,

(1 )

zi (0, x) = ηi (x) − ηi (x) = 0.

(2.15)

Since u i = qi ( w i ), q i ( w i ) = ( D i (u i ))−1 and

q

i ( w i ) = − D i (u i )−2

d  dw i





 −3

D i (u i ) = − D i (u i )

D i (u i )

we see that



(1 ) 

¯i q i w



   D i (ξi )  (1) ¯ i − w (i 1) − q i w (i 1) = − w 3 ( D i (ξi )) (1)

where ξi ≡ ξi (t , x) is an intermediate value between u¯ i that



(1)

(1)

¯ i ) and u i ≡ qi ( w

(1)

≡ qi ( w i ). This implies

(1 ) 

  ¯ (i 1) /∂ t − q i w (i 1) ∂ w (i 1) /∂ t ∂w  (1)  (1)    (1 )    ¯i ¯ i /∂ t − ∂ w (1) /∂ t + q i w ¯ i − q i w (i 1) ∂ w (i 1) /∂ t ∂w = q i w

(1 )

 (1 )  (1 ) D i (ξi ) ∂ w i ¯ i ∂ zi /∂ t − = q i w zi . ( D i (ξi ))3 ∂ t

¯i q i w

(2.16)

Using this relation in the first inequality in (2.15) yields



(1 ) 

¯i q i w

(1 )

(1 )

∂ zi /∂ t − L i zi

(1 )

+ γi z i

 0 in Q T

where



 

3

γi ≡ γi (t , x) ≡ − D i (ξi )∂ w (i 1) /∂ t / D i (ξi ) .

(2.17)

 uˆ i  δ ∗ > 0 or D i (0) > 0 that γi is bounded on Q T . By Lemma 2.1 of [28] ¯ (i 1)  w (i 1) for the case i = n0 , . . . , N. The above result shows that w ¯ (i 1) we have zi  0 which gives w (1) (0) (1) (1) (0) ¯i w ¯ i for i = 1, . . . , N. This ensures that and w i exist and possess the property w i  w i  w (1) ¯ (i 1) ), (u (i 1) , w (i 1) ) exist and satisfy the relation the first iterations (u¯ i , w (1)

It is clear from ξi  u i (1)



( 0)

( 0) 

ui , w i

 ( 1 ) ( 1 )   ( 1 ) ( 1 )   ( 0) ( 0)  ¯ i  u¯ i , w ¯i ,  u i , w i  u¯ i , w

i = 1, . . . , N .

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C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

Assume, by induction, that



(m−1)

ui

(m−1) 

, wi

 (m) (m)   (m) (m)   (m−1) (m−1)  ¯i ¯i  ui , w i  u¯ i , w  u¯ i , ,w (m)

for some m > 1. Then the function zi



(m)

¯i =w

(m+1)

¯i −w

i = 1, . . . , N

(2.18)

satisfies the relation

(m) 

 (m+1)  (m+1) (m) ¯ (i m) /∂ t − q i w ¯i ¯i ∂w ∂w /∂ t − L i zi             = F i t , x, u¯ (i m−1) , u¯ (m−1) a , u(m−1) b − F i t , x, u¯ (i m) , u¯ (m) a , u(m) b  0,

¯i q i w

i

(m)

zi

∗

i

i

i

∗

(t , x) = hi (t , x) − hi (t , x) = 0,

∂ z(i m) /∂ ν + c i(2) z(i m)             = G i t , x, u¯ (i m−1) , u¯ (m−1) c , u(m−1) d − G i t , x, u¯ (i m) , u¯ (m) c , u(m) d  0, i

(m)

zi

(m)

where u¯ i (m)

¯i (w

i

i

i

(0, x) = ηi (x) − ηi (x) = 0

¯ (i m) ) and u (i m) = qi ( w (i m) ). By the relation in (2.16) with ( w ¯ (i 1) , w (i 1) ) replaced by = qi ( w (1) ) the same argument as that for zi leads to

(m+1)

¯i ,w





¯ (m) ∂ zi q i w

(m)

(m)

/∂ t − L i zi

(m) (m)

+ γi

zi

 0 ( i = 1, . . . , N )

γi(m) ≡ γi(m) (t , x) is given by (2.17) with ξi replaced by some intermediate value ξi(m) between ¯ (i m+1) . A sim¯ (i m+1) . Since γi(m) is bounded we conclude that z(i m)  0. This yields w ¯ (i m)  w and w

where (m)

¯i w

(m)

(m+1)

(m+1)

(m+1)

¯i ilar argument gives w i  w i and w  wi . This shows that (2.18) holds when m is replaced by m + 1. The conclusion of the lemma follows by the induction principle. 2 For the uniqueness of the solution of (2.6) we prepare the following. Lemma 2.3. If (u∗ , w∗ ) is a solution of (2.6) in S then it satisfies the relation













¯ (m) , w ¯ (m) , u(m) , w(m)  u∗ , w∗  u

m = 1, 2, . . . .

(2.19)

Proof. Consider Problem (2.12) with P (t , x) = P¯ (0) (t , x), H (t , x) = h∗i (t , x) and i = 1, . . . , n0 − 1. Since ˆ,w ˆ )  (u∗ , w∗ )  (u˜ , w ˜ ), the components (u ∗i , w ∗i ) of (u∗ , w∗ ) for i = 1, . . . , n0 − 1 by hypothesis, (u satisfy the relation







   , u∗ b i in Q T   ( 0)  F i u˜ i , [u˜ ]ai , [uˆ ]bi = P¯ i (t , x) 

q i w ∗i ∂ w ∗i /∂ t − L i w ∗i = F i u ∗i , u∗



ai

w ∗i (t , x) = h∗i (t , x)

on S T

w ∗i (0, x) = ηi (x)

in Ω

(i = 1, . . . , n0 − 1).

This implies that w ∗i is a lower solution of (2.12). On the other hand, by the iteration pro(1)

¯i cess (2.11), w

is the solution of (2.12) (with P i = P¯ i

(0)

and H i = h∗i ) which is also an upper

¯ (i 1) )  (u ∗i , w ∗i )  w ∗i . This leads to (u¯ (i 1) , w ∗ for i = 1, . . . , n0 − 1. A similar argument using P i (t , x) = P i (t , x) and H i (t , x) = h i (t , x) in (2.12) gives (1)

¯i solution we concluded from Lemma 2.1 of [30] that w

(0)

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

(m−1)

(u ∗i , w ∗i )  (u i , w i ). Assume by induction, that (u i some m > 1. Then by using the property (1)



(m−1)

F i ui

(1)

(m−1)

, wi

1525

(m−1)

)  (u ∗i , w ∗i )  (u¯ i

(m−1)

¯i ,w

) for

           (m−1)  (m−1)   (m−1)    , u(m−1) a , u¯ (m−1) b  F i u ∗i , u∗ a , u∗ b  F i u¯ i , u¯ , u a b i

i

i

i

i

i

an induction argument leads to the relation



(m)

ui

     ¯ (i m) for i = 1, . . . , n0 − 1. , w (i m)  u ∗i , w ∗i  u¯ (i m) , w

˜ ¯ (i 1) − w ∗i . By (2.11), (2.9) and uˆ  u∗  u, To show the above relation for i = n0 , . . . , N, we let z∗i = w we have



(1 ) 

        ¯ (i 1) /∂ t − q i w ∗i ∂ w ∗i /∂ t − L i z∗i = F i u¯ (i 0) , u¯ (0) a , u(0) b ∂w i i  ∗  ∗  ∗  − F i ui , u a , u b  0 i i (i = n0 , . . . , N ).      ∗  ∗  ∗    (2 ) ∗ ( 0) ∗ ( 0 ) ( 0 ) −  ∂ zi /∂ ν + c i zi = G i u¯ i , u¯ , u G u , u , u 0 i i c d c d

¯i q i w

i

i

i

i

z∗i (0, x) = ηi (x) − ηi (x) = 0 (1)

By the same reasoning as that in the proof for zi

(1)

¯i in Lemma 2.2, we obtain w

 w ∗i which yields

¯ i )  (u ∗ , w ∗ ). A similar argument gives (u (i 1) , w (i 1) )  (u ∗i , w ∗i ). It follows by an induction (u¯ i , w (m) (m) (m) ¯ (i m) ) for every m and i = n0 , . . . , N. This proves the argument that (u i , w i )  (u ∗i , w ∗i )  (u¯ i , w (1)

(1)

relation (2.19).

2

Using the results of the above lemmas we now prove the existence and uniqueness of a classical solution to the system (2.6).

˜,w ˜ ), (uˆ , w ˆ ) be a pair of coupled upper and lower solutions of (2.6), and let HypotheTheorem 2.1. Let (u sis (H1 ) be satisfied. Then (i) Problem (2.6) has a unique solution (u∗ , w∗ ) in S × S , ¯ (m) , w ¯ (m) }, {u(m) , w(m) } governed by (2.11) converge monotonically to (u∗ , w∗ ), and (ii) the sequences {u (iii) these sequences possess the monotone property:

      ˆ )  u(m) , w(m)  u(m+1) , w(m+1)  u∗ , w∗ (uˆ , w     ¯ (m+1)  u¯ (m) , w ¯ (m)  (u˜ , w ˜ ), m = 1, 2, . . . .  u¯ (m+1) , w

(2.20)

Proof. In view of (2.13) the pointwise limits





¯ (m) , w ¯ (m) = (u¯ , w ¯ ), lim u

m→∞





lim u(m) , w(m) = (u, w)

(2.21)

m→∞

exist and satisfy the relation









¯,w ¯ )  u¯ (m) , w ¯ (m) , u(m) , w(m)  (u, w)  (u

m = 1, 2, . . . .

It has been shown in [30] (for i = 1, . . . , n0 − 1) and in [28] (for i = n0 , . . . , N) that given any subdo(m )

main Q T of Q T with Q T ⊂ Q T there exists a subsequence {u i

(m )

, wi

(m )

} such that ((u i

(m )

)t , ( w i

)t ),

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C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

(m )

((u i

(m )

)xk , ( w i (m )

(m )

)xk ) and ((u i

(m )

(m )

)xk xl , ( w i

)xk xl ) (k, l = 1, . . . , n) are uniformly convergent in Q T ,

(m )







¯ (i m ) ) or (u (i m ) , w (i m ) ). Since these sequences converge , w i ) stands for either (u¯ i , w ¯ i ) and (u i , w i ), respectively, we see that (u¯ i , w ¯ i ) and (u i , w i ) are in C 1,2 ( Q T ) × C 1,2 ( Q T ) and to (u¯ i , w where (u i

satisfy the equations









¯ i = f i t , x, u¯ i , [u¯ ]ai , [u]bi , ¯ i )∂ w ¯ i /∂ t − L i w q i ( w ( 0)

¯ ]bi , q i ( w i )∂ w i /∂ t − L i w i = f i t , x, u i , [u]ai , [u ( 0)

(2.22)

¯ i ) and (u i , w i ) where L i is given by (2.8) with c i = 0. The arbitrariness of Q T implies that (u¯ i , w ¯ i ) and (u i , w i ) satisfy (2.22) in the whole domain Q T . It has also been shown in [28,30] that (u¯ i , w are continuous up to the boundary S T and satisfy the boundary–initial conditions (0)

(1)

¯ i = w i = h∗i w





(i = 1, . . . , n0 − 1),

¯ i /∂ ν = g i t , x, u¯ i , [u¯ ]ci , [u]di ∂w   (i = n0 , . . . , N ), ∂ w i /∂ ν = g i t , x, u i , [u]ci , [u¯ ]di ¯ i (0, x) = w i (0, x) = ηi (x) w

(i = 1, . . . , N ).

(2.23)

¯ i ) and u i = qi ( w i ). Hence to show the It is obvious from the last relation in (2.11) that u¯ i = qi ( w ¯ i = w i for every i = 1, . . . , N. existence and uniqueness of a solution to (2.6) it suffices to show that w ¯ i − w i and zi = e −kt w i where k is a positive constant to be chosen. Then w i  0 and Let w i = w by (2.22) and (2.23),





¯ i )∂ w ¯ i /∂ t − q i ( w i )∂ w i /∂ t − L i zi e −kt q i ( w =e

  −kt



( 0)



¯ ]ai , [u]bi − f i t , x, u i , [u]ai , [u¯ ]bi f i t , x, u¯ i , [u

zi (t , x) = 0

∂ zi /∂ ν

 

= e −kt g i t , x, u¯ i , [u¯ ]ci , [u]di







(i = 1, . . . , N ),

≡ Ji

− g i t , x, u i , [u]ci , [u¯ ]di



(i = 1, . . . , n0 − 1), ≡

zi (0, x) = 0

J∗ i

(i = n0 , . . . , N ), (i = 1, . . . , N ).

(1)

(2.24)

(1)

¯ i , w i ) replaced by ( w ¯ i , w i ) we have By the relation (2.16) with ( w







¯ i )∂ w ¯ i /∂ t − q i ( w i )∂ w i /∂ t = e −kt q i ( w ¯ i )∂ w i /∂ t + γi w i e −kt q i ( w



¯ i )∂ zi /∂ t + kzi + γi zi = q i ( w

(2.25)

¯ i and w i . where γi ≡ γi (t , x) is given by (2.17) with ξi ≡ ξi (t , x) some intermediate value between w Moreover, by the mean-value theorem and the relation dq i /dw i = 1/ D i (u i ), we have

 

         ¯) b , q(w) b − f i t , x, qi ( w i ), q(w) a , q(w i i i 

ai bi ∂ f i dq j ∂ f i dqi ∂ f i dqi = e −kt wi + wj − wj ∂ u i dw i ∂ u j dw j ∂ u j dw j 



¯ i ), q(w ¯) J i = e −kt f i t , x, qi ( w

j =1

= c ii zi +

ai j =1

ci j z j −

bi j =1

ci j z j

ai

j =1

( i = 1, . . . , N )

(2.26)

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

1527

and similarly,

J i∗ = c ii∗ zi +

ci

c i∗j z j −

j =1

di

c i∗j z j

(i = n0 , . . . , N ),

j =1

where

∂ fi (ξ j )/ D j (ξ j ) (i , j = 1, . . . , N ), ∂u j   ∂ gi  ∗  ξ / D j ξ ∗j ( i = n 0 , . . . , N , j = 1, . . . , N ) c i∗j ≡ c i∗j (t , x) ≡ ∂u j j c i j ≡ c i j (t , x) =

¯ j and w j . It is clear from the mixed quasiand ξ j and ξ ∗j are some intermediate values between w monotone property of f(·, u), g(·, u) that c i j (t , x)  0 for j = 1, . . . , ai ,

c i j (t , x)  0 for j = 1, . . . , b i

(i = 1, . . . , N ),

c ∗ (t , x)  0

c ∗ (t , x)  0

(i = n 0 , . . . , N )

ij

for j = 1, . . . , c i ,

ij

for j = 1, . . . , di

(2.27)

and these functions are bounded on Q T if D j (ξ j ) = 0 and D j (ξ ∗j ) = 0. Using the relations (2.25)–(2.27) in (2.24) yields

¯ i )∂ zi /∂ t − L i zi + (k + γi − c ii ) zi = q i ( w ( 0)

N

|c i j | z j (i = 1, . . . , N ),

j =i

zi (t , x) = 0

∂ zi /∂ ν = c ii∗ zi +

(i = 1, . . . , n0 − 1), N  ∗ c  z j

(i = n0 , . . . , N ),

ij

j =i

zi (0, x) = 0

(i = 1, . . . , N ).

(2.28)

¯ We show that there exists a constant k¯ such that zi (t , x) = 0 on Q T for all i when k  k. Consider the case i = 1, . . . , n0 − 1 where zi (t , x) = 0 on S T and zi (0, x) = 0 in Ω . If zi = 0 then there exists (t¯, x¯ ) ∈ Q T such that zi (t¯, x¯ ) > 0. We claim that zi (t , x) > 0 for all (t , x) ∈ Q T . To see this we observe that if zi (t 0 , x0 ) = 0 for some (t 0 , x0 ) ∈ Q T then there would exist a subdomain Q T with

Q T ⊂ Q T such that zi (t 0 , x0 ) is a minimum of zi in Q T . In view of (2.28) and z j (t , x)  0 for all j we have

¯ i )∂ zi /∂ t − L i zi + (k + γi − c ii ) zi  0 in Q T . q i ( w ( 0)

ˆ i (t , x) > 0 and ξi (t , x)  uˆ i (t , x) > 0 on Q T the functions q i ( w ¯ i ), ¯ i (t , x)  w Since by the relations w

γi and c ii are all bounded on Q T , we conclude from the maximal principle that zi (t , x) cannot have an interior minimum in Q T . This contradiction shows that zi (t , x) > 0 in Q T . Let, for each k, (tk , xk ) be a maximum point of zi (t , x) in Q T . Then (tk , xk ) ∈ Q T and (∂ zi /∂ t )(tk , xk )  0,

(∇ zi )(tk , xk ) = 0,

 ∇ 2 zi (tk , xk )  0.



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This implies that

(k + γi − c ii ) · zi (tk , xk ) 

N

|c i j | z j (tk , xk ),

i = 1, . . . , n 0 − 1,

(2.29)

j =i

where γi = γi (tk , xk ) and c i j = c i j (tk , xk ), i = 1, . . . , n0 − 1, j = 1, . . . , N. Let zi be the maximum norm of zi (t , x) in Q T and let K ∗ > 0 be any constant satisfying K ∗ < 1/ N. We show that

zi  K ∗ N

N

z j for i = 1, . . . , n0 − 1.

(2.30)

j =1

Since the sequence {tk , xk } is bounded, there exists a subsequence, denoted again by {tk , xk }, such that limt →∞ (tk , xk ) = (t ∗ , x∗ ). We consider the following two cases about the limit point (t ∗ , x∗ ): Case 1. Either t ∗ = 0 or x∗ ∈ S T . In this case zi (t ∗ , x∗ ) = 0. Hence for any ε > 0 there exists K¯ > 0 such that zi (tk , xk ) < ε for all k  K¯ . This implies that for a sufficiently small ε > 0, zi satisfies (2.30) if z j = 0 for at least one j = i. In the case of z j = 0 for all j = i then zi satisfies the equation

¯ i )∂ zi /∂ t − L i zi + (k + γi − c ii ) zi = 0 in Q T . q i ( w ( 0)

From the proof for the positivity of zi (t , x) we see that either zi (t , x) ≡ 0 or zi (t , x) > 0 in Q T . Since −zi (t , x) satisfies also the above equation and the same boundary and initial conditions, we must have zi (t , x) = 0 in Q T . In this situation, (2.30) is trivially satisfied. Case 2. (t ∗ , x∗ ) ∈ Q T . In this case, there exists a neighborhood N (t ∗ , x∗ ) of (t ∗ , x∗ ) in Q T such that (tk , xk ) ∈ N (t ∗ , x∗ ) for all large k. Since γi and c i j are bounded in N (t ∗ , x∗ ), independent of k, there exists K¯ such that k + γ > 0 and c¯ /(k + γ )  K ∗ < 1/ N for all k  K¯ , where γ is a lower bound of γi (tk , xk ) and c¯ is an upper bound of c i j (tk , xk ) in N (t ∗ , x∗ ). In view of (2.29) we have

 −1

zi (tk , xk )  (k + γ )

|c ii | zi +

N

|c i j | z j  K ∗ N

j =i

N

z j for i = 1, . . . , n0 − 1.

j =1

This proves the relation (2.30). We next consider the case i = n0 , . . . , N where zi satisfies the nonlinear boundary condition in (2.28). Since by the hypothesis uˆ i  δi∗ > 0 and D i (u¯ i )  D i (uˆ i ) > 0 we see that the functions

¯ i ), γi , c i j and c i∗j are all bounded on Q T . This implies that for each i = n0 , . . . , N, the problem q i ( w in (2.28) is a uniformly parabolic boundary problem. By the standard integral representation of the N N ∗ solution for linear parabolic boundary problems with j =1 |c i j | z j and j =1 |c i j | z j as internal and boundary sources, respectively, we obtain an upper bound of zi in the form zi (t , x)  K i t μ

N

z j t ,

(t , x) ∈ Q T (i = n0 , . . . , N )

j =1

where K i and

μ are positive constants with μ < 1 and    z j t ≡ max  z j (τ , x); 0  τ  t , x ∈ Ω

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1529

μ

(cf. [25, p. 488]). The above relation implies that for any t 1 satisfying K i t 1  K ∗

zi t  K ∗

N

z j t for (t , x) ∈ [0, t 1 ] × Ω, i = n0 , . . . , N ,

(2.31)

j =1

where K ∗ is the constant in (2.30). Addition of the inequalities in (2.30) and (2.31) leads to the estimate N

zi t  N K ∗

i =1

N

z j t for 0  t  t 1 .

j =1

Since N K ∗ < 1, the above inequality is possible only if zi = 0 for every i = 1, . . . , N. This shows that zi (t , x) cannot have a positive maximum in Q 1 = [0, t 1 ] × Ω and therefore zi (t , x) = 0 on Q 1 . Using the domain (t 1 , T ] × Ω and the initial condition zi (t 1 , x) = 0 in (2.28), a similar argument shows that there exists t 2 > t 1 such that zi (t , x) = 0 on [t 1 , t 2 ] × Ω . A ladder argument in t leads ¯ ) = (u, w). ¯ i = w i for every i = 1, . . . , N which shows that (u¯ , w to zi (t , x) = 0 on Q T . This yields w ¯,w ¯ ) (or (u, w)) is a solution of (2.6). Since by Lemma 2.3 and (2.21) every By (2.22) and (2.23), (u ¯,w ¯ ) we conclude that (u¯ , w ¯) solution (u∗ , w∗ ) of (2.6) in S satisfies the relation (u, w)  (u∗ , w∗ )  (u is the unique solution of (2.6) in S . This proves the results in (i) and (ii). The conclusion in (iii) is a direct consequence of (ii) and Lemma 2.2. This completes the proof of the theorem. 2 Since by (2.5), q i ( w i )∂ w i cess (2.11) can be written as (m)

(m)

(m)

/∂ t = ∂ u i

(m)

/∂ t and ∇ w i

(m)

= D i (u i

(m)

)∇ u i

the iteration pro-

 (m)  (m)   (m)     (m)  (m)  (m) + b∗i · D i u¯ i ∇ u¯ i + γi I i u¯ i ∂ u¯ i /∂ t − ∇ · a∗i D i u¯ i ∇ u¯ i  (m−1)   (m−1)  (m−1)   (m−1)   + f i t , x, u¯ i = γi I i u¯ i , u¯ , u ai bi      (m)  (i = 1, . . . , N ),   (m) (m) (m) (m)  (m)  ∗ ∗ + bi · D i u i ∇ u i + γi I i u i ∂ u i /∂ t − ∇ · ai D i u i ∇ u i         = γi I i u (i m−1) + f i t , x, u (i m−1) , u(m−1) a , u¯ (m−1) b i

(m)

u¯ i

i

(m)

(t , x) = u i (t , x) = hi (t , x)   (m)  (m) ∂ u¯ i /∂ ν + c i(2) I i u¯ (i m) D i u¯ i  (2)  (m−1)  (m−1)  (m−1)   (m−1)   + g i t , x, u¯ i = c i I i u¯ i , u¯ , u ci di  (m)  (m)   (2 ) (m) ∂ u i /∂ ν + c i I i u i D i ui  (2)  (m−1)  (m−1)  (m−1)   (m−1)   + g i t , x, u i = ci I i ui , u , u¯ c d 

i

(m)

u¯ i

(m)

(0, x) = u i

(0, x) = ψi (x)

(i = 1, . . . , n0 − 1),

(i = n0 , . . . , N ),

i

(i = 1, . . . , N ).

(2.32)

As a consequence of Theorem 2.1 we have the following conclusion for the original system (2.1).

˜ uˆ be a pair of coupled upper and lower solutions of (2.1), and let Hypothesis (H1 ) be Theorem 2.2. Let u, satisfied. Then (i) Problem (2.1) has a unique solution u∗ (t , x) in S , ¯ (m) }, {u(m) } governed by (2.32) exist and converge to the solution u∗ (t , x), and (ii) the sequences {u

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(iii) these sequences possess the monotone property

ˆ  u(m)  u(m+1)  u∗  u¯ (m+1)  u¯ (m)  u˜ on Q T . u

(2.33)

When b i = di = 0 for all i, the functions f(·, u), g(·, u) are quasimonotone nondecreasing and the ˜ uˆ become ordered upper and lower solutions. By an application of Theorem 2.2 we have pair u, the following conclusion which is an extension of the existence result in [28,30] to mixed type of boundary conditions and will be needed in a later section.

˜ uˆ be a pair of ordered upper Theorem 2.3. Let f(·, u), g(·, u) be quasimonotone nondecreasing in S , and let u, and lower solutions of (2.1). Assume that Hypothesis (H1 ) is satisfied. Then all the conclusions in (i)–(iii) of ¯ (m) }, {u(m) } are independent to each other and are governed Theorem 2.2 hold true. Moreover, the sequences {u by (2.32) with b i = di = 0 for all i. The discussion for the system (2.1) can be easily extended to the parabolic-ordinary system (1.3) where the functions f (·, u), g (·, u) are mixed quasimonotone. As in Problem (2.1) we write Problem (1.3) in the form

      ∂ u i /∂ t − ∇ · a∗i D i (u i )∇ u i + b∗i · D i (u i )∇ u i = f i t , x, u i , [u]ai , [u]bi (i = 1, . . . , n1 ),   (i = n1 + 1, . . . , N ), ∂ u i /∂ t = f i t , x, u i , [u]ai , [u]bi u i (t , x) = h i (t , x)



D i (u i )∂ u i /∂ ν = g i t , x, u i , [u]c i , [u]di

(i = 1, . . . , n0 − 1),



(i = n0 , . . . , n1 ),

u i (0, x) = ψi (x)

(i = 1, . . . , N ).

(2.34)

The above system may be written in the form of (2.1) with D i (u i ) = 0 and without the boundary condition for i = n1 + 1, . . . , N. The definition of coupled (or ordered) upper and lower solutions for (2.34) is given by Definition 2.1 except with the requirements for i = n1 + 1, . . . , N be replaced by

  ∂ u˜ i /∂ t  f i t , x, u˜ i , [u˜ ]ai , [uˆ ]bi   (i = n1 + 1, . . . , N ). ∂ uˆ i /∂ t  f i t , x, uˆ i , [uˆ ]ai , [u˜ ]bi Moreover, the iteration process for the system (2.34) is the same as that in (2.32) where the equations for i = n1 + 1, . . . , N are replaced by

 (m) (1) (m) (1) (m−1) (m−1)  (m−1)   (m−1)   ∂ u¯ i /∂ t + c i u¯ i = c i u¯ i + f i t , x, u¯ i , u¯ , u ai bi (i = n1 + 1, . . . , N ).  (m) (1) (m) (1) (m−1) (m−1)  (m−1)   (m−1)   ¯ ∂ u i /∂ t + c i u i = c i u i + f i t , x, u i , u , u a b i

i

(2.35) It is seen from the discussion for the corresponding transformed problem (2.6) (with D i (u i ) = 0 for ¯ (m) }, {u(m) } are well-defined. By the same reasoning as that i = n1 + 1, . . . , N) that the sequences {u in the discussion for Problems (2.6) and (2.1) we have the following result which is useful in the application of many concrete physical and biological problems.

˜ uˆ be a pair of coupled upper and lower solutions of (2.34), and let Hypothesis (H1 ) be Theorem 2.4. Let u, satisfied. Then (i) Problem (2.34) has a unique solution u∗ in S ,

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

1531

¯ (m) }, {u(m) } governed by (2.32), where the equations for i = n1 + 1, . . . , N are replaced (ii) the sequences {u by (2.35), are well-defined and converge to u∗ , and (iii) the above sequences and u∗ satisfy the relation (2.33). Proof. The proof is a slight modification of the argument in the proof of Theorem 2.1 with D i (0) = 0 (and without the boundary condition) for i = n1 + 1, . . . , N. Details are omitted. 2 3. Quasilinear elliptic system In this section we investigate the existence problem for the elliptic system (1.2). As for Problem (1.1) we write the system (1.2) in the equivalent form

      −∇ · a∗i D i (u i )∇ u i + b∗i · D i (u i )∇ u i = f i x, u i , [us ]ai , [us ]bi (i = 1, . . . , N ), u i (x) = h i (x)



D i (u i )∂ u i /∂ ν = g i x, u i , [us ]c i , [us ]di

(i = 1, . . . , n0 − 1),



(i = n 0 , . . . , N )

(3.1)

where for convenience, we write the components of us by u i ≡ u i (x), i = 1, . . . , N.

˜ s ≡ (u˜ 1 , . . . , u˜ N ), uˆ s ≡ (uˆ 1 , . . . , uˆ N ) in C (Ω) ∩ C 2 (Ω) are called Definition 3.1. A pair of functions u ˜ s  uˆ s > 0 in Ω and if they satisfy the relation coupled upper and lower solutions of (3.1) if u

      −∇ · a∗i D i (u˜ i )∇ u˜ i + b∗i · D i (u˜ i )∇ u˜ i  f i x, u˜ i , [u˜ s ]ai , [uˆ s ]bi     (i = 1, . . . , N ),   −∇ · a∗i D i (uˆ i )∇ uˆ i + b∗i · D i (uˆ i )∇ uˆ i  f i x, uˆ i , [uˆ s ]ai , [u˜ s ]bi u˜ i (x)  h i (x)  uˆ i (x)









˜ s ]ci , [uˆ s ]di D i (u˜ i )∂ u˜ i /∂ ν  g i x, u˜ i , [u ˆ s ]ci , [u˜ s ]di D i (uˆ i )∂ uˆ i /∂ ν  g i x, uˆ i , [u

(i = 1, . . . , n0 − 1), (i = n0 , . . . , N ).

(3.2)

˜ s and uˆ s are also coupled upper and lower solutions It is clear from the above definition that u ˆ s  ψ  u˜ s , where ψ ≡ (ψ1 , . . . , ψ N ). We assume that a pair of coupled upper and of (2.1) whenever u ˜ s , uˆ s exist and set lower solutions u



S ∗i ≡ u i ∈ C α (Ω); uˆ i  u i  u˜ i



 S ≡ us ∈ C α (Ω); uˆ s  us  u˜ s . ∗



(i = 1, . . . , N ), (3.3)

For the elliptic system (3.1) we make the following hypothesis:

(H2 )

(i) For each i = 1, . . . , N, the functions a∗i ≡ a∗i (x), b∗i ≡ b∗i (x), h i ≡ h i (x), f i ≡ f i (x, u) and g i ≡ g i (x, u) are independent of t and satisfy the conditions in (H1 )(i) with S replaced by S ∗ . (ii) D i (u i ) satisfies Hypothesis (H1 )(ii) with S i and S replaced, respectively, by S ∗i and S ∗ . (iii) f i (·, u) and g i (·, u) are mixed quasimonotone in S ∗ and satisfy the condition (2.4) with (1) (1) (2) (2) c i ≡ c i (x)  0, c i ≡ c i (x)  0.

ˆs > 0 In the above hypothesis we do not require f i (x, 0)  0 for i = 1, . . . , n0 − 1. The requirement u ˆ s must in Definition 3.1 is only for x ∈ Ω , and it may vanish on ∂Ω . In fact, the components uˆ i of u be zero on ∂Ω if h i (x) = 0. To construct monotone sequences as that in (2.11) we transform Problem (3.1) into a semilinear system by the transformation w i = I i [u i ] in (2.5). This leads to the system

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    −∇ · a∗i ∇ w i + b∗i · ∇ w i = f i x, u i , [us ]ai , [us ]bi (i = 1, . . . , N ), w i (x) = h∗i (x)



∂ w i /∂ ν = g i x, u i , [us ]ci , [us ]di

(i = 1, . . . , n0 − 1),



(i = n0 , . . . , N ),

u i = qi ( w i )

( i = 1, . . . , N )

(3.4)

where h∗i (x) = I i [h i (x)]. The above system can also be obtained from (2.6) without the time-derivative terms and the initial condition. It is to be noted that upon replacing us by q(ws ) in the functions f i (·, us ) and g i (·, us ), Problem (3.4) becomes a system of semilinear equations in ws ≡ ( w 1 , . . . , w N ). However, since dw i /du i = ( D i (u i ))−1 which is unbounded if D i (0) = 0 the function f i (·, q(ws )) or g i (·, q(ws )) may not be Lipschitz continuous with respect to ws . For the convenience of constructing monotone sequences we use the form (3.4) and introduce the following definition of upper and lower solutions.

˜ s, w ˜ s ), (uˆ s , w ˆ s ) in C (Ω) ∩ C 2 (Ω) are called coupled upper and Definition 3.2. A pair of functions (u ˜ s, w ˜ s )  (uˆ s , w ˆ s ) > (0, 0) in Ω and if they satisfy the relation lower solutions of (3.4) if (u

    ˜ i + b∗i · ∇ w ˜ i  f i x, u˜ i , [u˜ s ]ai , [uˆ s ]bi −∇ · a∗i ∇ w    (i = 1, . . . , N ),  ˆ i + b∗i · ∇ w ˆ i  f i x, uˆ i , [uˆ s ]ai , [u˜ s ]bi −∇ · a∗i ∇ w ˆ i (x) ˜ i (x)  h∗i (x)  w w 

(i = 1, . . . , n0 − 1),



˜ i /∂ ν  g i x, u˜ i , [u˜ s ]ci , [uˆ s ]di ∂w   ˆ i /∂ ν  g i x, uˆ i , [uˆ s ]ci , [u˜ s ]di ∂w

(i = n0 , . . . , N ),

˜ i ), u˜ i  qi ( w

(i = 1, . . . , N ).

ˆ i) uˆ i  qi ( w

(3.5)

˜ s , uˆ s are coupled upper and lower solutions of (3.1) then It is obvious from Definition 3.1 that if u ˜ s, w ˜ s ) ≡ (u˜ s , I [u˜ s ]), (uˆ s , w ˆ s ) ≡ (uˆ s , I [uˆ s ]) are coupled upper and lower solutions of (3.4). the pair (u ˜ s, w ˜ s ), (uˆ s , w ˆ s ) we set For a given pair of (u

  ˆ s )  (us , ws )  (u˜ s , w ˜ s) . S ∗ × S ∗ ≡ (us , ws ) ∈ C α (Ω) × C α (Ω); (uˆ s , w (0)

(0)

(0)

(0)

¯s ,w ¯ s ) = (u˜ s , w ˜ s ) and (us , ws ) = (uˆ s , w ˆ s ) as a pair of coupled initial iterations we Using (u (m)

¯s construct two sequences {u

(m)

(m)

¯ s }, {us , w(sm) } from the linear iteration process ,w

 (m−1)  (m−1)   (m−1)   = F i x, u¯ i , u¯ s , us ai bi     (m−1)   (m) (m−1) (m−1) − L i w i = F i x, u i , us , u¯ s a b (m)

¯i −L i w

i

(i = 1, . . . , N ),

i

(x) = h∗i (x) (i = 1, . . . , n0 − 1),  (m−1)  (m−1)   (m−1)   (m) (2) (m) ¯ i = G i x, u¯ i ¯ i /∂ ν + c i w ∂w , u¯ s , us ci di  (m−1)  (m−1)   (m−1)   (i = n0 , . . . , N ), (m) (2) (m) ∂ w i /∂ ν + c i w i = G i x, u i , us , u¯ s ci di     (m) (m) (m) (m) ¯i , ( i = 1, . . . , N ) u¯ i = qi w u i = qi w i (m)

¯i w

(m)

(x) = w i

(2)

(2)

(3.6)

where L i , F i and G i are given by (2.8) (with respect to the functions in (3.1)) and c i ≡ c i (x) ≡ 0 on ∂Ω . Since for each i and each m the above problem is a linear elliptic boundary problem with Dirichlet boundary condition for i = 1, . . . , n0 − 1 and Robin boundary condition for i = n0 , . . . , N, ¯ (sm) , w ¯ (sm) }, {u(sm) , w(sm) } are well-defined. We show that these sequences converge the sequences {u ¯ s, w ¯ s ), (us , ws ) in the sense that they satisfy the equations monotonically to a pair of quasisolutions (u

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

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    ¯ i + b∗i · ∇ w ¯ i = f i x, u¯ i , [u¯ s ]ai , [us ]bi −∇ · a∗i ∇ w    (i = 1, . . . , N ),    −∇ · a∗i ∇ w i + b∗i · ∇ w i = f i x, u i , us a , [u¯ s ]bi i

¯ i (x) = w

w i (x) = h∗i (x)



(i = 1, . . . , n0 − 1),



¯ i /∂ ν = g i x, u¯ i , [u¯ s ]ci , [us ]di ∂w   ∂ w i /∂ ν = g i x, u i , [us ]ci , [u¯ s ]di

(i = n0 , . . . , N ),

¯ i ), u¯ i = qi ( w

(i = 1, . . . , N ).

u i = qi ( w i )

(3.7)

Specifically, we have the following results:

˜ s, w ˜ s ), (uˆ s , w ˆ s ) be a pair of coupled upper and lower solutions of (3.4), and let HypotheTheorem 3.1. Let (u sis (H2 ) be satisfied. Then the following statements hold true: (m)

(m)

(m)

(m)

¯s ,w ¯ s }, {us , ws } governed by (3.6) converge monotonically to a pair of quasiso(i) The sequences {u ¯ s, w ¯ s ), (us , ws ) that satisfy (3.7) and possess the monotone property lutions (u

    ˆ s )  u(sm) , w(sm)  u(sm+1) , w(sm+1)  (us , ws )  (u¯ s , w ¯ s) (uˆ s , w  (m+1) (m+1)   (m) (m)  ¯s ¯s ˜ s ), m = 1, 2, . . . .  u¯ s , w  (u˜ s , w  u¯ s ,w

(3.8)

(ii) Every solution (u∗s , w∗s ) of (3.4) in S ∗ × S ∗ satisfies the relation

  ¯ s ) in Ω (us , ws )  u∗s , w∗s  (u¯ s , w

(3.9)

and there exists at least one such a solution if either D i (0) > 0 or uˆ i  δ0 > 0 for i = n0 , . . . , N. ¯ s, w ¯ s ) (≡ (u∗s , w∗s )) then (u∗s , w∗s ) is the unique solution of (3.4) in S ∗ × S ∗ . (iii) If (us , ws ) = (u Proof. It is easily seen from the argument in the proof of Lemma 2.2 (without the time-derivative ¯ (sm) , w ¯ (sm) } is nonincreasing, {u(sm) , w(sm) } is nonterms and the initial condition) that the sequence {u (m)

decreasing, and they satisfy the relation (us pointwise limits



(m)

¯s lim u

m→∞

(m)

(m)

(m)

¯ s ) for every m. This implies that the , ws )  (u¯ s , w

 ¯ (sm) = (u¯ s , w ¯ s ), ,w



(m)

lim us

m→∞

 , w(sm) = (us , ws )

(3.10)

exist and satisfy (3.8). To show that these limits are quasisolutions of (3.4) we consider the linear boundary problem (m−1)

− L i w i = P¯ i

for any fixed m > 1, where

(x) in Ω,

αi ∂ w i /∂ ν + βi w i = H¯ i(m−1) (x) on ∂Ω

(3.11)

αi = 0, βi = 1 for i = 1, . . . , n0 − 1 and αi = 1, βi = c i(2) for i = n0 , . . . , N.

(m−1) (m−1) The functions P¯ i (x), H¯ i (x) are given by (2.14) with respect to the functions in (3.1) which (m)

¯i are independent of t. Since by (3.6), w i = w (m−1)

¯ H i

(m−1) is the unique solution of (3.11) and P¯ i (x),

(x) are uniformly bounded in Ω , the standard regularity argument for linear boundary prob(m) ¯ i ∈ C 2+α (Ω) (cf. [25, p. 102 and } converges in C 2 (Ω) to some function w (m−1) (m−1) (m−1) (m−1) ¯ ¯ p. 158]). Upon replacing ( P i (x), H i (x)) by ( P i (x), H i (x)) the same reasoning shows (m) (m) ¯ i ) ≡ u¯ i that { w i } converges in C 2 (Ω) to some function w i ∈ C 2+α (Ω). This implies that u¯ i → qi ( w ¯i lems shows that { w

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(m)

¯ i ) and (u i , w i ) satisfy and u i → qi ( w i ) ≡ u i as m → ∞. Letting m → ∞ in (3.6) shows that (u¯ i , w the equation in (3.7). This proves the result in (i). (1) (0) ˆs  (ii) Let w∗s ≡ ( w ∗1 , . . . , w ∗N ), u∗s = (u ∗1 , . . . , u ∗N ) and zi = w ∗i − w i , i = 1, . . . , N. Since u0 = u ˜ s = u¯ s u∗s  u

(0)

we see from (2.8), (2.9) and (3.6) that

            − L i zi = F i x, u ∗i , u∗s a , u∗s b − F i x, u (i 0) , u(s0) a , u¯ (s0) b  0 i

zi = h i − h i = 0 (2 )

∂ zi /∂ ν + c i zi =



i

i







 

G i x, u ∗i , u∗s c , u∗s d i i



(i = 1, . . . , N ),

i

( 0)



( 0) 

− G i x, u i , us



ci

( 0) 

, u¯ s

di



(i = 1, . . . , n0 − 1),  0 (i = n0 , . . . , N ).

Hence whether i = 1, . . . , n0 − 1 or i = n0 , . . . , N the positivity lemma for linear boundary prob(1) (1) (1) lems ensures that zi  0, or equivalently, w ∗i  w i . In view of u ∗i = I i [ w ∗i ] and u i = I i [ w i ]

we have u ∗i  u i . This shows that (u∗s , w∗s )  (us , ws ). A similar argument using the prop(1)

(1)

(1)

¯s ,w ¯ s ). It follows by an induction argument that erty of an upper solution gives (u∗s , w∗s )  (u (1)

(1)

¯ s ) for every m = 1, 2, . . . . Letting m → ∞ and using the rela(us , ws )  (u∗s , w∗s )  (u¯ s , w ¯ s, w ¯ s ). tion (3.10) yields (us , ws )  (u∗s , w∗s )  (u To show the existence of a solution we write (m)

(m)

(m)

(m)

  F i (x, u) = F i x, q(w) ≡ F i∗ (x, w), 



G i (x, u) = G i x, q(w) ≡ G ∗i (x, w),

i = 1, . . . , N

(3.12)

and consider the linear boundary problem

− L i w i = F i∗ (x, v)

(x ∈ Ω),

αi ∂ w i /∂ ν + βi w i =

H ∗ (x, v) i

(x ∈ ∂Ω),

i = 1, . . . , N

(3.13)

where v is an arbitrary given function in C 1+α (Ω), and

H i∗ (x, v) = h∗i (x)

α i = 0,

β i = 1,

α i = 1,

βi ≡ c i(2) (x),

for i = 1, . . . , n0 − 1,

H i∗ (x, v) = G ∗i (x, v)

for i = n0 , . . . , N .

By (3.12) and the hypothesis D i (0) > 0 or uˆ i (x)  δ0 > 0 for i = n0 , . . . , N, the functions F i∗ (x, w)

ˆ s, w ˜ s  where w ˆ s = I [uˆ s ] and w ˜ s = I [u˜ s ]. Hence for any and G ∗i (x, w) are C 1 -functions of w ∈ S ∗ ≡ w

v ∈ S ∗ ∩ C 1+α (Ω) the above problem has a unique solution w i ∈ C 2+α (Ω) for each i. Denote the solution operator A by w = Av where w = ( w 1 , . . . , w N ). Then A maps C 1+α (Ω) into C 2+α (Ω) and ˜ s. ˆs ww is a compact operator in C 1+α (Ω) (cf. [25, p. 500]). We show that w ˜ s and w ˆs vw ˆ s = I [uˆ s ], w ˜ s = I [u˜ s ] ˆ i . In view of w Let u = q(v) (or v = I [u]) and let zi = w i − w ˆ s  u  u˜ s and F i∗ (x, v) = F i (x, u), G ∗i (x, v) = G i (x, u). By (2.13), (3.5) and the mixed quasiwe have u monotone property of F i (x, u) and G i (x, u),

    ˆ i  F i x, u i , [u]ai , [u]bi − F i x, uˆ i , [uˆ s ]ai , [u˜ s ]bi  0 (i = 1, . . . , N ), L i zi = F i∗ (x, v) − L (1) w ˆi 0 zi = h∗i − w





ˆi ˆ i /∂ ν + c i w ∂ zi /∂ ν + c i zi = G ∗ (x, v) − ∂ w      G i x, u i , [u]ci , [u]di − G i x, uˆ i , [uˆ s ]ci , [u˜ ]di  0 (2 )

(2 )

(i = 1, . . . , n0 − 1), (i = n0 , . . . , N ).

ˆ i for each i = 1, . . . , N. A similar argument gives This implies that zi  0, or equivalently, w i  w ˜ s . Moreover, by the standard imbedding theorem ˆ s  Av  w ˜ i for every i. This shows that w wi  w

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

1535

for linear boundary problems there exists a constant K ∗ such that Av 1+α  K ∗ (cf. [25, p. 501]). Define a closed convex set B ∗ of C 1+α (Ω) by

  ˆs vw ˜ s , v 1+α  K ∗ . B∗ ≡ v ∈ C 1+α (Ω); w Then the above conclusions show that A maps B ∗ into itself. Since A is precompact on B ∗ and Av ∈ C 2+α (Ω) for every v ∈ B∗ the Schauder fixed point theorem ensures that A has a fixed point ˜ s which ensures that (u∗s , w∗s ) = (q(w∗s ), w∗s ) is a ˆ s  w∗s  w w∗s ∈ B ∗ . This implies w∗s = Aw∗s and w solution of (3.4). (iii) The result in (iii) is an immediate consequence of the conclusion in (ii). 2 It is obvious from (3.7) that if f(·, u) and g(·, u) are quasimonotone nondecreasing in u, that is, if ¯ s, w ¯ s ) and (us , ws ) are solutions of (3.4). The result in (ii) implies b i = di = 0 for all i, then both (u that they are the respective maximal and minimal solutions in S ∗ × S ∗ . This observation leads to the following conclusion which is a direct extension of a result in [28,30] to mixed Dirichlet and Neumann–Robin boundary conditions. Theorem 3.2. Let the conditions in Theorem 3.1 be satisfied. If the functions f(x, u), g(x, u) are quasimonotone ¯ s, w ¯ s ) and (us , ws ) are true solutions of (3.4). In fact, they are the nondecreasing in u for u ∈ S ∗ then (u respective maximal and minimal solutions in S ∗ × S ∗ . (m)

¯i In the iteration process (3.6) if w then it is reduced to the form

(m)

and w i

(m)

are replaced by I i [u¯ i

] and I i [u (i m) ], respectively,

 (m)  (m)     (m)  (m)  (1)  (m)  + b∗i · D i u¯ i ∇ u¯ i + c i I i u¯ i −∇ · a∗i D i u¯ i ∇ u¯ i  (m−1)  (m−1)   (m−1)   , u¯ s , us = F i x, u¯ i ai bi  ∗  (m)  (m)    (m)  (m)  (1)  (m)  ∗ + bi · D i u i ∇ u i + ci I i ui −∇ · ai D i u i ∇ u i       = F i x, u (i m−1) , u(sm−1) a , u¯ (sm−1) b i

(m)

u¯ i

(i = 1, . . . , N ),

i

(m)

(x) = u i (x) = hi (x) (i = 1, . . . , n0 − 1),  (m−1)  (m−1)   (m−1)   (m)  (m) (2)  (m)  ∂ u¯ i /∂ ν + c i I i u¯ i = G i x, u¯ i D i u¯ i , u¯ s , us ci di  (m)  (m)  (m−1)  (m−1)   (m−1)   (i = n0 , . . . , N ) (2)  (m)  ∂ u i /∂ ν + c i I i u i = G i x, u i D i ui , us , u¯ s (3.14) c d 

i

(0)

(0)

(0)

i

(0)

(m)

¯s ,w ¯ s ) = (u˜ s , I [u˜ s ]) and (us , ws ) = (uˆ s , I [uˆ s ]). By Theorem 3.1, the sequences {u¯ s where (u (m)

},

{us } are well-defined and converge to a pair of quasisolutions u¯ s , us that satisfy the equations

      −∇ · a∗i D i (u¯ i )∇ u¯ i + b∗i · D i (u¯ i )∇ u¯ i = f i x, u¯ i , [u¯ s ]ai , [us ]bi      (i = 1, . . . , N ),  −∇ · a∗i D i (u i )∇ u i + b∗i · D i (u i )∇ u i = f i x, u i , [us ]ai , [u¯ s ]bi u¯ i (x) = u i (x) = h i (x)









¯ s ]ci , [us ]di D i (u¯ i )∂ u¯ i /∂ ν = g i x, u¯ i , [u ¯ s ]di D i (u i )∂ u i /∂ ν = g i x, u i , [us ]c i , [u

(i = 1, . . . , n0 − 1), (i = n0 , . . . , N ).

(3.15)

This observation leads to the following results for the original system (3.1).

˜ s , uˆ s be a pair of coupled upper and lower solutions of (3.1), and let Hypothesis (H2 ) be Theorem 3.3. Let u satisfied. Then the following statements hold true:

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(m)

(m)

¯ s }, {us } governed by (3.14) are well-defined and converge to a pair of quasisolutions (i) The sequences {u ¯ s , us that satisfy (3.15) and the relation u (m)

ˆ s  us u

(m+1)

 us

(m+1)

 us  u¯ s  u¯ s

(m)

 u¯ s

 u˜ s ,

m = 1, 2, . . . .

(3.16)

¯ s and there exists at least one such (ii) Every solution u∗s of (3.1) in S ∗ satisfies the relation us  u∗s  u a solution if either D i (0) > 0 or uˆ i  δ0 > 0 for i = n0 , . . . , N. ¯ s = us (≡ u∗s ) then u∗s is the unique solution of (3.1) in S ∗ . (iii) If u ¯ s and us are the respective maximal and minimal solutions of (3.1) in S ∗ . (iv) If b i = di = 0 for all i then u The above discussion for the elliptic system (3.1) can be extended to the steady-state problem of the parabolic-ordinary system (1.3) (or (2.34)) which is given by

      −∇ · a∗i D i (u i )∇ u i + b∗i · D i (u i )∇ u i = f i x, u i , [us ]ai , [us ]bi (i = 1, . . . , n1 ),   f i x, u i , [us ]ai , [us ]bi = 0 (i = n1 + 1, . . . , N ), u i (x) = h i (x)



D i (u i )∂ u i /∂ ν = g i x, u i , [us ]c i , [us ]di

(i = 1, . . . , n0 − 1),



(i = n0 , . . . , n1 ).

(3.17)

It is obvious that Problem (3.17) may be considered as a special case of (3.1) with D i (u i ) = 0 and without the boundary condition for i = n1 + 1, . . . , N. This implies that the definition of coupled upper and lower solutions for (3.17) is given by Definition 3.1 where the differential inequalities for i = n1 + 1, . . . , N are replaced by









˜ s ]ai , [uˆ s ]bi  0  f i x, uˆ i , [uˆ s ]ai , [u˜ s ]bi , f i x, u˜ i , [u

i = n 1 + 1, . . . , N (m)

¯s (and without the boundary inequalities for i = n1 + 1, . . . , N). The iteration process for {u is given by (3.14) except that for i = n1 + 1, . . . , N they are replaced by

 (m−1)  (m−1)   (m−1)   (1)  (m−1)  = c i I i u¯ i + f i x, u¯ i , , u¯ s , us ai bi         (1)  (m)  = c i(1) I i u (i m−1) + f i x, u (i m−1) , u(sm−1) a , u¯ (sm−1) b . ci I i ui (1 )



(m)

} , {u s }

(m) 

c i I i u¯ i

i

i

(3.18)

As a consequence of Theorem 3.3 we have the following

˜ s , uˆ s be a pair of coupled upper and lower solutions of (3.17), and let Hypothesis (H2 ) be Theorem 3.4. Let u satisfied where D i (u i ) = 0 for i = n1 + 1, . . . , N. Then all the conclusions in (i), (ii), (iii), and (iv) of Theo¯ (sm) }, {u(sm) } are governed by (3.14) and (3.18). rem 3.3 hold true for Problem (3.17) where the sequences {u Proof. The proof follows from the arguments in the proofs of Theorem 3.1 to Theorem 3.3 with D i (u i ) = 0 for i = n1 + 1, . . . , N. Details are omitted. 2 4. Global attractor and stability To investigate the asymptotic behavior of the solution of (2.1) (or (1.1)) we assume that the elliptic ˜ s , uˆ s and Hypothesis (H2 ) system (3.1) (or (1.2)) has a pair of coupled upper and lower solutions u ¯ s , us such that holds. In view of Theorem 3.3 the elliptic system (3.1) has a pair of quasisolutions u ˆ s  us  u¯ s  u˜ s . Since u˜ s and uˆ s are also coupled upper and lower solutions of (2.1) for any ψ u ˆ s  ψ  u˜ s , we see from Theorem 2.2 that a unique global solution u∗ (t , x) to (2.1) exists satisfying u and satisfies the relation

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

ˆ s (x)  u∗ (t , x)  u˜ s (x) u

for all t > 0, x ∈ Ω

1537

(4.1)

provided that it holds at t = 0 and the additional condition

f i (x, 0)  0 for i = 1, . . . , n0 − 1 (x ∈ Ω)

(4.2)

ˆ s , u˜ s  between uˆ s and u˜ s is an invariant set in (H1 )(iii) is satisfied. This implies that the sector S ∗ ≡ u ¯ s  between the of the parabolic system (2.1). In the following discussion we show that the sector us , u ¯ s is a global attractor of Problem (2.1) (relative to S ∗ ), and if u¯ s = us (≡ u∗s ) quasisolutions us and u then u∗s is a global attractor. To achieve this goal we assume that one of the following conditions is satisfied when 1 < n0 < N + 1 (mixed boundary condition). (H3 ) Either (a) D i (0) > 0 for i = 1, . . . , n0 − 1, or (b) h i (x)  h0 > 0 on ∂Ω for i = 1, . . . , n0 − 1, or (c) g i (t , x, u) = D i (u i ) g i (t , x, u i ) and

(∂ g i /∂ u i )(t , x, u i )  0 for u i ∈ S i , i = n0 , . . . , N . The condition in (H3 )(c) includes the linear Neumann–Robin boundary condition

∂ u i /∂ ν + βi (x)u i = hi (x) for i = n0 , . . . , N , where βi (x)  0 on ∂Ω . Notice that the above condition is a special case of the boundary condition in (1.1) with





g i (t , x, u) = D i (u i ) h i (x) − βi (x)u i ,

i = n0 , . . . , N .

We first consider the case where f(x, u) and g(x, u) are quasimonotone nondecreasing in u for u ∈ S ∗ . In the following lemmas we always assume that in addition to (H2 ) the following hypothesis is satisfied for the case of n0 = N + 1 and 1 < n0 < N + 1 (but not for n0 = 1).

(H4 ) (a) Condition (4.2) holds if n0 = N + 1 (Dirichlet boundary condition). (b) Condition (4.2) and Hypothesis (H3 ) hold if 1 < n0 < N + 1. Consider the time-dependent problem (2.1) and the corresponding steady-state problem (3.1) for the case b i = di = 0 for all i = 1, . . . , N. Assume that Problem (3.1) has a pair of ordered upper and ˜ s , uˆ s . By Theorem 3.3, Problem (3.1) has a maximal solution u¯ s and a minimal solower solutions u ˆ s  us  u¯ s  u˜ s . Let u¯ (t , x) and u(t , x) be the solutions of (2.1) with their lution us that satisfy u ¯ (0, x) = u˜ s (x) and u(0, x) = uˆ s (x). The following lemma gives the monorespective initial conditions u tone property (in t) of these solutions.

¯ (t , x) Lemma 4.1. Let Hypotheses (H2 ) and (H4 ) be satisfied. Then the solution u(t , x) is nondecreasing in t, u ¯ (t , x) on Q T . Moreover, for any ψ ∈ uˆ s , u˜ s  the corresponding solution is nonincreasing in t, and u(t , x)  u u∗ (t , x) of (2.1) satisfies the relation

¯ (t , x) u(t , x)  u∗ (t , x)  u

on Q T .

(4.3)

Proof. The result in the lemma is known for the case n0 = 1 and n0 = N + 1 (cf. [28,30]). We prove ˆ s (x) the general case where 1 < n0 < N + 1. Consider the solution u(t , x). It is obvious from u(0, x) = u that u(t , x) satisfies the relation (4.1). Let z(t , x) = u(t + δ, x) − u(t , x) for any constant δ > 0, and

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let g i∗ (x, u) = g i (t , u)/ D i (u i ). By the quasimonotone nondecreasing property of g(x, u), the function g∗ (x, u) ≡ ( g 1∗ (x, u), . . . , g ∗N (x, u)) is also quasimonotone nondecreasing in u for u ∈ S ∗ . In view of (2.1) and the mean-value theorem we have

∂ zi /∂ t − A i (t , x) zi + Bi (t , x) · ∇ zi + C i (t , x) zi N     = f i x, u(t + δ, x) − f i x, u(t , x) = ci j z j

(i = 1, . . . , N ),

j =1

zi (t , x) = 0

(i = 1, . . . , n0 − 1),

N     ∂ zi /∂ ν = g i∗ x, u(t + δ, x) − g i∗ x, u(t , x) = βi j z j (i = n0 , . . . , N ), j =1

zi (0, x) ≡ zi ,0 (x) = u i (δ, x) − uˆ i (x)

( i = 1, . . . , N )

(4.4)

where





A i (t , x) = a∗i (x) D i u i (t + δ, x) ,









Bi (t , x) = −a∗i (x)∇ D i u i (t + δ, x) − a∗i (x) D i (θi )∇ u i (t , x) + D i u (t + δ, x) bi (x),









C i (t , x) = −a∗i (x)∇ · D i (θi )∇ u i (t , x) + D i (θi )bi · ∇ u i (t , x)

(4.5)

and θi is some intermediate value between u i (t , x) and u i (t + δ, x) (cf. [28,30]). The functions c i j ≡ c i j (t , x) and βi j ≡ βi j (t , x) in (4.4) are given by

ci j ≡

∂ fi (x, ξ ), ∂u j

βi j =

∂ g i∗   x, ξ , ∂u j

i , j = 1, . . . , N

where ξ ≡ ξ (t , x) and ξ ≡ ξ (t , x) are some intermediate values between u(t , x) and u(t + δ, x). It is clear that c i j are bounded in Q T and from the hypothesis uˆ i  δi∗ > 0 for i = n0 , . . . , N that βi j are bounded on S T . Moreover the quasimonotone nondecreasing property of f(x, u) and g∗ (x, u) implies that c i j  0 on Q T and βi j  0 on S T for j = i. We show that u i (t + δ, x)  u i (t , x) on Q T for every i. By the relation u i (t , x)  uˆ i (t , x) > 0 in Q T we see from D i (u i ) > 0 for u i > 0 that A i (t , x)  A 0 for some constant A 0 > 0 if either D i (0) > 0 or u i (t , x)  h0 > 0 on S T . This follows from condition (H3 )(a) and condition (H3 )(b), respectively, when i = 1, . . . , n0 − 1 because u i (t , x) = h i (x)  h0 . Moreover, the hypothesis uˆ i (t , x)  δi∗ > 0 for i = n0 , . . . , N ensures that A i (t , x)  A 0 for i = n0 , . . . , N. This implies that Problem (4.4) is a uniformly linear parabolic system for the unknown function (z1 , . . . , z N ). Since by (4.1), zi ,0 = u i (δ, x) − uˆ i (x)  0 and the functions c i j and βi j are nonnegative for all j = i we conclude from a result in [25] (see p. 564) that zi (t , x)  0 for every i = 1, . . . , N. This gives u i (t + δ, x)  u i (t , x) when either condition (a) or condition (b) in Hypothesis (H3 ) holds. We next show that u i (t + δ, x)  u i (t , x) if condition (c) in (H3 ) is satisfied. Let Z i = e −kt zi for a positive constant k to be chosen. Then Z i satisfies the equations

∂ Z i /∂ t − A i  Z i + Bi · ∇ Z i − (k + C i ) Z i =

N

ci j Z j

(i = 1, . . . , N ),

j =1

Z i (t , x) = 0

(i = 1, . . . , n0 − 1),

∂ Z i /∂ ν = βi Z i

(i = n0 , . . . , N ),

Z i (0, x) = zi ,0 (x) ≡ u i (δ, x) − uˆ i (x)

( i = 1, . . . , N )

(4.6)

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

1539

where βi ≡ βi (t , x) = (∂ g i /∂ u i )(x, ξi ) for some intermediate value ξi ≡ ξi (t , x) between u i (t , x) and u i (t + δ, x). By Hypothesis (H3 )(c), βi (t , x)  0 on S T . Assume by contradiction that Z i is not nonnegative. Then there would exist an i and a point (t i , xi ) in Q T such that Z i (t i , xi ) is the smallest negative minimum of Z j (t , x) for all j. By the boundary and initial conditions in (4.6), where βi (t i , xi ) Z i (t i , xi )  0 we have (t i , xi ) ∈ Q T . This implies that (∂ Z i /∂ t )(t i , xi )  0, ( Z i )(t i , xi )  0, and (∇ Z i )(t i , xi ) = 0. Using this relation and the nonnegative property of c i j for j = i in (4.6) yield

(k + C i − c ii ) Z i (t i , xi ) 

N

c i j Z j (t i , xi )

j =i

 

N

c i j Z i (t i , xi ).

j =i

N

The above relation leads to a contradiction if we choose k > −C i + j =1 c i j . This shows that Z i (t , x)  0 which is equivalent to u i (t + δ, x)  u i (t , x), i = 1, . . . , N. Hence under any one of the conditions in (H3 ) the relation u(t + δ, x)  u(t , x) on Q T holds. The arbitrariness of δ > 0 im¯ (t , x) is simiplies that u(t , x) is nondecreasing in t. The proof for the nonincreasing property of u lar. To show the relation (4.3) we let zi (t , x) = u ∗i (t , x) − u i (t , x) and observe that zi (0, x) = u ∗i (0, x) − uˆ i (x)  0, i = 1, . . . , N. Upon replacing u i (t + δ, x) by u ∗i (t , x) in the above discussion we obtain ¯ (t , x). u∗ (t , x)  u(t , x). A similar argument using the relation u ∗i (0, x)  u˜ i (x) leads to u∗ (t , x)  u This proves the lemma. 2 In view of Lemma 4.1 and the arbitrariness of T > 0 the pointwise limits

¯ (t , x) = u¯ (x) lim u

t →∞

and

lim u(t , x) = u(x)

t →∞

(x ∈ Ω)

(4.7)

ˆ (x)  u(x)  u¯ (x)  u˜ (x). We show that u¯ (x) = u¯ s (x) and u(x) = us (x). exist and satisfy the relation u ¯ (x), u(x) in (4.7) coincide with the respective maximal and minimal solutions u¯ s (x) Lemma 4.2. The limits u and us (x). Moreover, the solution u∗ (t , x) in Lemma 4.1 satisfies the relation

¯ s (x) us (x)  u∗ (t , x)  u

as t → ∞ (x ∈ Ω).

(4.8)

¯ (x) and u(x) Proof. It is easy to show by the same regularity argument as that in [28,30] that both u ¯ (x) = u¯ s (x) and u(x) = us (x). Consider the are solutions of (3.1) (with b i = di = 0). We show that u ¯ (x). Since u¯ (0, x) = u˜ s (x)  u¯ s (x) and u¯ s (x) may be considered as a solution of (2.1) with case for u ψ(x) = u¯ s (x) we see from Lemma 4.1 (with u(t , x) = u¯ s (x)) that u¯ s (x)  u¯ (t , x). Letting t → ∞ and ¯ s (x)  u¯ (x). The maximal property of u¯ s (x) in S ∗ ensures u¯ (x) = u¯ s (x). using the relation (4.7) yield u A similar argument by considering us (x) as a solution of (2.1) gives u(x) = us (x). Finally, the relation in (4.8) follows from (4.3) and the above conclusion. 2 In view of Lemmas 4.1 and 4.2 we have the following results for Problem (2.1) where b i = di = 0 for all i.

˜ s , uˆ s be a pair of ordered upper and lower solutions of (3.1), and let u¯ s , us be the respective Theorem 4.1. Let u maximal and minimal solutions of (3.1) in S ∗ . Assume that f(x, u) and g(x, u) are quasimonotone nondecreasˆ s  ψ  u˜ s ing in u for u ∈ S ∗ , and that Hypotheses (H2 ) and (H4 ) are satisfied. Then for any ψ satisfying u the corresponding solution u∗ (t , x) of (2.1) possesses the convergence property

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lim u∗ (t , x) =

t →∞



us (x)

ˆ s  ψ  us if u

¯ s (x) u

¯ s  ψ  u˜ s if u

(x ∈ Ω).

(4.9)

¯ s (x) = us (x) (≡ u∗s (x)), then Moreover, if u lim u∗ (t , x) = u∗s (x)

t →∞

(x ∈ Ω).

(4.10)

Proof. The above result is known for the case n0 = 1 and n0 = N + 1, and we prove it for the case ˆ s (x) and consider 1 < n0 < N + 1 (cf. [28,30]). Let u(t , x) be the solution of (2.1) with u(0, x) = u ¯ (t , x) ≡ us (x) as the solution of (2.1) with u¯ (0, x) = us (x). Since uˆ s (x)  us (x), Lemma 4.1 implies u ˆ s  ψ  us , the solution that u(t , x)  us (x). The same lemma ensures that for any ψ satisfying u u∗ (t , x) satisfies u(t , x)  u∗ (t , x)  us (x). Letting t → ∞ and using Lemma 4.2 and (4.7) we obtain limt →∞ u∗ (t , x) = us (x). This proves the first relation in (4.9). The proof for the second relation is ¯ s = us ≡ u∗s then the relation (4.10) is a direct consequence of (4.8). This similar. It is obvious that if u proves the theorem. 2 To investigate the asymptotic behavior of the solution of (1.2) for the general case b i + di = 0 for some i we extend the problem to a coupled system of 2N equations with quasimonotone nondecreasing functions. To achieve this, we introduce a new function v ≡ v 1 , . . . , v N  by letting v = M − u, ˜ s on Ω . Define functions where M = M 1 , . . . , M N  is a positive constant satisfying M > u

¯ i ( v i ) = D i ( M i − v i ), D (1 )

h¯ i = M i − h i ,

ψ¯ i = M i − ψi ,



(1 )



v i

¯ i (s) ds, D

0



F i (x, u, v) = c i I i [u i ] + f i x, u i , [u]ai , [M − v]bi ,

¯I i ( v i ) =



(2 ) (1 ) F i (x, u, v) = c i ¯I i [ v i ] − f i x, M i − v i , [M − v]ai , [u]bi , (1 )



(2 )



G i (x, u, v) = c i I i [u i ] + g i x, u i , [u]c i , [M − v]di ,





(2 ) (2 ) G i (x, u, v) = c i ¯I i [ v i ] − g i x, M i − v i , [M − v]c i , [u]di , (1)

(1)

(2)

(4.11)

(2)

where c i = c i (x) and c i = c i (x) are the positive functions satisfying (2.4). Then the extended problem is given by the coupled system

    ∂ u i /∂ t − ∇ · a∗i D i (u i )∇ u i + b∗i · D i (u i )∇ u i + c i(1) I i [u i ]   = F i(1) x, u i , [u]ai , [M − v]bi     ∂ v i /∂ t − ∇ · a∗i D¯ i ( v i )∇ v i + b∗i · D¯ i ( v i )∇ v i + c i(1) ¯I i [ v i ]  (2 )  = F i x, M i − v i , [M − v]ai , [u]bi v i (t , x) = h¯ i (x)

u i (t , x) = h i (x), (2 )

(1 ) 



D i (u i )∂ u i /∂ ν + c i I i [u i ] = G i

x, u i , [u]c i , [M − v]di

¯ i ( v i )∂ v i /∂ ν + c I i [ v i ] = G D i i

x, M i − v i , [M − v]c i , [u]di

(2 ) ¯

u i (0, x) = ψi (x),

(2 ) 

v i (0, x) = ψ¯ i (x)

Similarly, the elliptic system (3.1) is extended to the form

(i = 1, . . . , N ),

(i = 1, . . . , n0 − 1),  (i = n0 , . . . , N ), (i = 1, . . . , N ).

(4.12)

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

    −∇ · a∗i D i (u i )∇ u i + b∗i · D i (u i )∇ u i + c i(1) I i [u i ]   = F i(1) x, u i , [u]ai , [M − v]bi     −∇ · a∗i D¯ i ( v i )∇ v i + b∗i · D¯ i ( v i )∇ v i + c i(1) ¯I i [ v i ]  (2 )  = F i x, M i − v i , [M − v]ai , [u]bi u i (x) = h i (x),

v i (x) = h¯ i (x) (2 )

(1 ) 

(i = 1, . . . , N ),

(i = 1, . . . , n0 − 1),



D i (u i )∂ u i /∂ ν + c i I i [u i ] = G i

x, u i , [u]c i , [M − v]di

¯ i ( v i )∂ v i /∂ ν + c (2) ¯I i [ v i ] = G D i i

x, M i − v i , [M − v]c i , [u]di

(2 ) 

1541

 (i = n0 , . . . , N ).

(4.13)

It is obvious that u(t , x) is a solution of (2.1) if and only if (u, v) ≡ (u, M − v) is a solution of (4.12); and us (x) is a solution of (3.1) if and only if (us , vs ) ≡ (us , M − us ) is a solution of (4.13). Moreover, by (2.4) and the mixed quasimonotone property of f(·, u) and g(·, u) in S ∗ , the 2N-vector functions

 (1 )  (1 ) (2 ) (2 ) F (x, u, v) = F 1 (x, u, v), . . . , F N (x, u, v), F 1 (x, u, v), . . . , F N (x, u, v) ,  (1 )  (1 ) (2 ) (2 ) G (x, u, v) = G 1 (x, u, v), . . . , G N (x, u, v), G 1 (x, u, v), . . . , G N (x, u, v)

(4.14)

∗ , where are nondecreasing in (u, v) for (u, v) ∈ S ∗ × S M

  ∗ SM ≡ v ∈ C α (Ω); M − u˜ s  v  M − uˆ s .

(4.15)

˜ s , v˜ s ) ∈ C 2 (Ω)∩ Because of the nondecreasing property of F (·, u, v) and G (·, u, v), we call a function (u C (Ω) an upper solution of (4.13) if it satisfies (4.13) with all the equality sign “=” replaced by the ˆ s , vˆ s ) is called a lower solution if it satisfies (4.13) with “=” replaced inequality sign “”. Similarly, (u ˜ s , v˜ s ), (uˆ s , vˆ s ) are said to be ordered if (u˜ s , v˜ s )  (uˆ s , vˆ s ). In terms of u˜ s , uˆ s we by “”. The pair (u have the following lemma. ˜ s , v˜ s ) ≡ (u˜ s , M − uˆ s ), (uˆ s , vˆ s ) ≡ (uˆ s , M − u˜ s ) are ordered upper and lower solutions Lemma 4.3. The pair (u ˜ s and uˆ s are coupled upper and lower solutions of (3.1). of (4.13) if and only if u ˜ s , uˆ s be coupled upper and lower solutions of (3.1), and let v˜ s = M − uˆ s , vˆ s = M − u˜ s . Proof. Let u Since by (4.11)

¯ i ( v˜ i ) = D¯ i ( M i − uˆ i ) = D i (uˆ i ), D

¯ i ( vˆ i ) = D¯ i ( M i − u˜ i ) = D i (u˜ i ), D

Definition 3.1 implies that

    −∇ · a∗i D¯ i ( v˜ i )∇ v˜ i + b∗i · D¯ i ( v˜ i )∇ v˜ i + c i(1) ¯I i ( v˜ i )      = − −∇ · a∗i D i (uˆ i )∇ uˆ i − b∗i · D i (uˆ i )∇ uˆ i + c i(1) ¯I i [ v˜ i ]   (1 )  − f i x, uˆ i , [uˆ s ]ai , [u˜ s ]bi + c i ¯I i [ v˜ i ]   (1 ) (2 ) = c i ¯I i [ v˜ i ] − f i x, M i − v˜ i , [M − v˜ ]ai , [u˜ s ]bi = F i (x, u˜ s , v˜ s )

(i = 1, . . . , N ),

v˜ i (x) = M i − uˆ i (x)  M i − h i (x) = h¯ i (x)

(i = 1, . . . , n0 − 1),

(2 ) ¯

¯ i ( v˜ i )∂ v˜ i /∂ ν + c I i [ v˜ i ] D i

  (2 ) (2 ) = D i (uˆ i )(−∂ uˆ i /∂ ν ) + c i ¯I i [ v˜ i ]  − g i x, uˆ i , [uˆ s ]ci , [u˜ s ]di + c i ¯I i [ v˜ i ] (i = n0 , . . . , N ).   (2 ) (2 ) = c i ¯I i [ v˜ i ] − g i x, M i − v˜ i , [M − v˜ ]ci , [u˜ s ]di = G i (x, u˜ s , v˜ s )

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˜ s , v˜ s ) satisfies the remaining inequalities in (4.13) where the equality sign It is easy to verify that (u ˜ s , v˜ s ) is an upper solution of (4.13). A similar “=” is replaced by the inequality sign “”. This shows (u ˆ s , vˆ s ) ≡ (uˆ s , M − u˜ s ) is a lower solution. The ordering relation (u˜ s , v˜ s )  (uˆ s , vˆ s ) argument shows that (u ˜ s  uˆ s . The proof for the converse is similar and is omitted. 2 follows from u In view of Lemma 4.3 we can apply Theorem 3.3 to the extended system (4.13) where the ∗ . Specifically, Theofunctions F (x, u, v) and G (x, u, v) in (4.14) are nondecreasing in (u, v) ∈ S ∗ × S M ¯ , v¯ ) and a minimal solution (u, v) such rem 3.3 ensures that Problem (4.13) has a maximal solution (u that

(uˆ s , M − u˜ s )  (u, v)  (u¯ , v¯ )  (u˜ s , M − uˆ s ). ¯ s and us The following lemma gives a relationship between these solutions and the quasisolutions u of (3.1). ˜ s , uˆ s be coupled upper and lower solutions of (3.1), and let u¯ s , us be the quasisolution in S ∗ . Lemma 4.4. Let u ¯ , v¯ ) = (u¯ s , M − us ) and (u, v) = (us , M − u¯ s ) are the maximal and minimal solutions of (4.13) in Then (u ∗. S ∗ × SM Proof. Consider the iteration process

 (m) (m)     (m)  (m)  (1)  (m)  + b∗i · D i u i ∇ u i + ci I i ui −∇ · a∗i D i u i ∇ u i   (1 )  (m−1)  (m−1)   , u , M − u(m−1) b = F i x, u i ai i   (i = 1, . . . , N ),  ∗  (m)  (m)    (m)  (m)  ∗ ¯ ¯ + bi · D i v i ∇ v i + c i(1) ¯I i v (i m) −∇ · ai D i v i ∇ v i       = F i(2) x, M i − v (i m−1) , M − v(m−1) a , u(m−1) b i

u i (x) = h i (x),



(m) 

i

v i (x) = h¯ i (x)

(m)

(2 )



(i = 1, . . . , n0 − 1),

(m) 

∂ u i /∂ ν + c i I i u i    (1 ) (m−1)  (m−1)   , u , M − v(m−1) d = G i x, u i ci i     ¯D i v (m) ∂ v (m) /∂ ν + c (2) ¯I i v (m) i i i i       = G (i 2) x, M i − v (i m−1) , M − v(m−1) c , u(m−1) d D i ui

i

(i = n 0 , . . . , N ) (4.16)

i

˜ s , M − uˆ s ) or (u(0) , v(0) ) = (uˆ s , M − u˜ s ). Since by Lemma 4.3, (u˜ s , M − uˆ s ) where either (u(0) , v(0) ) = (u ˆ s , M − u˜ s ) are ordered upper and lower solution of (4.13), Theorem 3.3 ensures that {u¯ (m) , v¯ (m) } and (u ¯ , v¯ ) while {u(m) , v(m) } converges monoconverges monotonically from above to a maximal solution (u ∗ . On the other hand, since the se∗ tonically from below to a minimal solution (u, v) in S × S M (m)

¯s quences {u

(m)

(m)

(m)

(m)

(m)

(m)

(m)

, v¯ s } ≡ {u¯ s , M − us }, and {us , vs } ≡ {us , M − u¯ s } satisfy also the equations ¯ (sm) } and {u(sm) } are the sequences governed by (3.14) (with u¯ (s0) = u˜ s , u(s0) = uˆ s ) the in (4.16), where {u ¯ (m) , v¯ (m) ) = (u¯ (sm) , M − u(sm) ) and uniqueness property of the iteration process in (4.16) implies that (u (m) (m) ( m) (m) (u , v ) = (us , M − u¯ s ) for every m. This implies that

 (m) (m)  = (u¯ s , M − us ), (u¯ , v¯ ) = lim u¯ s , M − us m→∞

 (m) (m)  (u, v) = lim us , M − u¯ s = (us , M − u¯ s ). m→∞

This proves the lemma.

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The results in the above lemmas for the extended system (4.13) yield the following conclusion for the original system (3.1).

˜ s , uˆ s be a pair of coupled upper and lower solutions of (3.1), and let Hypotheses (H2 ) Theorem 4.2. Let u ¯ s , us be the quasisolutions of (3.1) in S ∗ and u∗ (t , x) the solution of (2.1) with and (H4 ) be satisfied. Let also u ∗ an arbitrary ψ ∈ S . Then

¯ s (x) us (x)  u∗ (t , x)  u

as t → ∞ (x ∈ Ω).

(4.17)

¯ s (x) = us (x) (≡ u∗s (x)) then u∗s (x) is the unique solution of (3.1) in S ∗ and Moreover, if u lim u∗ (t , x) = u∗s (x)

(x ∈ Ω).

t →∞

(4.18)

¯ (t , x), v¯ (t , x)) and (u(t , x), v(t , x)) be the solutions of the extended system (4.12) with Proof. Let (u (u¯ (0, x), v¯ (0, x)) = (u˜ s (x), M − uˆ s (x)) and (u(0, x), v(0, x)) = (uˆ (x), M − u˜ s (x)). Since by Lemma 4.3 ˜ s , M − uˆ s ) and (uˆ s , M − u˜ s ) are ordered upper and lower solutions of (4.13) we see from the pair (u ¯ (t , x), v¯ (t , x)) converges monotonically from above Lemma 4.1 for the extended system (4.12) that (u ¯ (x), v¯ (x)) of (4.13) and (u(t , x), v(t , x)) converges monotonically from below to a maximal solution (u ¯ ∈ S∗ × S∗ to a minimal solution (u(x), v(x)) as t → ∞. Moreover, for arbitrary initial function (ψ, ψ) M the corresponding solution (u∗ (t , x), v∗ (t , x)) satisfies the relation













¯ (x), v¯ (x) u(x), v(x)  u∗ (t , x), v∗ (t , x)  u

as t → ∞.

¯ (x), v¯ (x)) = (u¯ s (x), M − us (x)) and (u(x), v(x)) = (us (x), M − u¯ s (x)) the However, by Lemma 4.4, (u ¯ s = us ≡ u∗s (x) then u∗s (x) is the unique above relation ensures that (4.17) holds. It is obvious that if u ∗ solution of (3.1) in S and (4.18) holds. 2 It is seen from Hypothesis (H4 ) that one of the conditions in (H3 ) is required only for the mixed boundary condition where 1 < n0 < N + 1. Hence as a special case of Theorem 4.2 we have the following conclusion which is a direct extension of a major result in [28,30] to mixed quasimonotone functions. Because of its usefulness in the application to specific problems, such as the model problems in the next section, we state it as a theorem.

˜ s , uˆ s be a pair of coupled upper and lower solutions of (3.1) for the cases n0 = 1 and Theorem 4.3. Let u ¯ s , us be the quasisolutions of (3.1) in S ∗ . Assume that Hypothesis (H2 ) is satisfied. Then n0 = N + 1, and let u all the conclusions in Theorem 4.2 hold true for the nonlinear boundary problem (2.1) where n0 = 1. The same is true for the Dirichlet boundary problem (2.1) where n0 = N + 1 if condition (4.2) is satisfied. Remark 4.1. (a) The conclusion in Theorem 4.1 implies that for quasimonotone nondecreasing func¯ s  between the maximal solution u¯ s and the minimal sotions f(·, u) and g(·, u) the sector us , u ¯ s and us are one-sided lution us is an attractor of the time-dependent problem (2.1). Moreover, u ˆ s , us  and u¯ s , u˜ s , respectively. For mixed quasimonoasymptotically stable with a stability region u ¯ s and us are quasisolutions, and by Theorem 4.2 the sector tone functions f(·, u), g(·, u), the pair u

us , u¯ s  is also an attractor of Problem (2.1). In both cases, if u¯ s = us ≡ u∗s then u∗s is a global attractor (relative to S ∗ ) of the time-dependent problem. (b) The results in Theorems 4.1 and 4.2 remain true for the parabolic-ordinary system (1.3) and its corresponding elliptic system (1.4). This can be shown by the above arguments with D i (u i ) = 0 and without the boundary condition for i = n1 + 1, . . . , N. Details are omitted. 5. Application to some model problems To give some applications of the results obtained in the previous sections we consider three model problems with porous medium type of diffusion which may be degenerate on the boundary of the

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domain. These model problems with constant diffusion coefficient have been extensively investigated in the area of mathematical biology, especially in relation to the dynamics of the time-dependent problem (cf. [8,17,25,44–46]). Our aim in the following discussion is to determine the large time behavior of the time-dependent solution in relation to the steady-state solutions or quasisolutions of the corresponding steady-state problem, including the existence and uniqueness of the time-dependent solution and the existence of solutions or quasisolutions of the steady-state problem. In each problem, the asymptotic behavior of the time-dependent solution may exhibit rather interesting distinct property when compared with the same problem with constant diffusion. 5.1. An extended logistic reaction–diffusion equation Consider the following extended porous medium type of logistic reaction–diffusion problem



ut − d0 um = u r a − bu s



(t > 0, x ∈ Ω),

α ∂ u /∂ ν + β u = 0

(t > 0, x ∈ ∂Ω),

u (0, x) = ψ(x)

(x ∈ Ω)

(5.1)

and its corresponding steady-state problem

  −d0 um = ur a − bu s (x ∈ Ω),

α ∂ u /∂ ν + β u = 0

(x ∈ ∂Ω)

(5.2)

where d0 , a, b, r, s and m are positive constants with m > 1 and either α = 0, β = 1 (Dirichlet condition) or α = 1, β  0 (Neumann–Robin condition). It is obvious that Problem (5.1) is a special case of (1.1) (respectively, (1.2)) with N = 1, u 1 = u, a∗1 = d0 , b∗1 = 0 and

D 1 (u 1 ) = mum−1 ,





f 1 (u 1 ) = u r a − bu s ,

h1 (x) = 0 if α = 0, β = 1

and

g 1 (u 1 ) = −β mum

if α1 = 1, β  0.

(5.3)

This implies that Hypotheses (H1 ) and (H2 ) are satisfied by the functions in (5.3). The above problem for the semilinear case m = 1 has been investigated by many investigators in the earlier literature especially for the case r = s = 1 (cf. [25]). It is well-known in the special case m = r = s = d0 = 1 that for any initial function ψ(x) > 0 in Ω the solution u (t , x) of (5.1) converges to the trivial solution u s = 0 of (5.2) if a  λ(1) and it converges to a unique positive solution u ∗s (x) of (5.2) if a > λ(1) , where λ(1) > 0 is the smallest eigenvalue of the eigenvalue problem

φ + λ(m) φ = 0 in Ω,

α ∂φ/∂ ν + mβφ = 0 on ∂Ω

(5.4)

corresponding to m = 1 (cf. [25]). This implies that if a  λ(1) then u s = 0 is a global attractor. However, this is no longer true if m > 1. In fact, we show that for the general problem (5.1) where m > max{1, r }, the steady-state problem (5.2) has a unique positive solution u ∗s (x) for any a > 0, and the solution u (t , x) of (5.1) converges to u ∗s (x) as t → ∞. To prove the above conclusion we observe that the boundary condition in (5.2) is equivalent to

α D 1 (u )∂ u /∂ ν + mβ um = 0 on ∂Ω for both cases α = 0, β = 1 and α = 1, β  0. In each case, the constant u˜ s = ρ is an upper solution of (5.2) if ρ  (a/b)1/s . To find a lower solution we let uˆ s = (δφm )1/m for a sufficiently small constant δ > 0, where φm is the (normalized) positive eigenfunction of (5.4) corresponding to the smallest

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ˆm ˆ s is a eigenvalue λ(m) . Indeed, since by (5.3), D 1 (uˆ s )∂ uˆ s /∂ ν = ∂ uˆ m s /∂ ν and u s = δφm we see that u lower solution of (5.2) if





d0 λ(m) (δφm )  (δφm )r /m a − b(δφm )s/m ,

α δ∂φm /∂ ν + mβ(δφm )  0. By (5.4) and the hypothesis r /m < 1 the above inequalities are satisfied by a sufficiently small δ > 0. This construction shows that the pair

u˜ s (x) = ρ

and



1/m

uˆ s (x) = δφm (x)

are ordered upper and lower solutions of (5.2). In view of φm (x) = 0 on ∂Ω when α = 0, β = 1 and φm (x) > 0 on ∂Ω when α = 1, β  0, the requirement uˆ s (x)  δ1∗ > 0 for the nonlinear boundary condition in (H1 ) is also satisfied. Hence by Theorem 3.3, Problem (5.2) has a maximal solution u¯ s (x) and a minimal solution u s (x) such that (δφm )1/m  u s (x)  u¯ s (x)  ρ . To show the uniqueness of the steady-state solution, we observe from (5.3) that

f (u ) I [u ]

=

u r (a − bu s ) um

=

a − bu s um−r

which is a decreasing function of u for 0  u  (a/b)1/s . By Theorem 3.3 of [26], u¯ s (x) = u s (x) (≡ u ∗s (x)) and u ∗s (x) is the unique solution of (5.2) in the sector uˆ s , u˜ s . Since ρ can be chosen arbitrarily large and δ arbitrarily small, u ∗s (x) is the unique positive solution. As an application of Theorem 4.1 (or Theorem 4.3) we have the following conclusion. Theorem 5.1. Let d0 , a, b, r, s and m be positive constants with m > max{1, r }, and let either α = 0, β = 1 or α = 1, β  0. Then Problem (5.2) has a unique positive solution u ∗s (x)  (a/b)1/s . Moreover, for any initial function ψ(x) satisfying (δφm )1/m  ψ  (a/b)1/s , where δ > 0 can be arbitrarily small, a unique positive solution u ∗ (t , x) to (5.1) exists and converges to u ∗s (x) as t → ∞. 5.2. A degenerate predator–prey model We next consider a predator–prey reaction–diffusion model with density-dependent diffusion and Dirichlet boundary condition in the form

ut − d1 um1 = u (a1 − b1 u − c 1 v ) v t − d2  v m2 = v (a2 + b2 u − c 2 v ) u (t , x) = v (t , x) = 0 u (0, x) = ψ1 (x),

(t > 0, x ∈ Ω), (t > 0, x ∈ ∂Ω),

v (0, x) = ψ2 (x)

(x ∈ Ω),

(5.5)

where ai , b i , c i , di and mi , i = 1, 2, are positive constants with mi > 1, (ψ1 (x), ψ2 (x)) > (0, 0) in Ω and (ψ1 (x), ψ2 (x)) = (0, 0) on ∂Ω . The corresponding steady-state problem is

−d1 um1 = u (a1 − b1 u − c 1 v ) −d2  v m2 = v (a2 + b2 u − c 2 v ) u (x) = v (x) = 0

(x ∈ Ω), (x ∈ ∂Ω).

(5.6)

It is obvious that Problem (5.5) is a special case of (1.1) with N = n0 = 2, (u 1 , u 2 ) = (u , v ), a∗i = di , b∗i = 0 and

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D 1 ( u 1 ) = m 1 u m 1 −1 ,

D 2 ( u 2 ) = m 2 v m 2 −1 ,

f 1 (u) = u (a1 − b1 u − c 1 v ),

h1 (x) = h2 (x) = 0,

f 2 (u) = v (a2 + b2 u − c 2 v ).

The above functions satisfy (4.2) and all the conditions in Hypothesis (H3 ). Problems (5.5) and (5.6) for the semilinear case m1 = m2 = 1 have been investigated extensively in the earlier literature (cf. [25]). It is known in the semilinear case m1 = m2 = 1 that for any (ψ1 , ψ2 ) > (0, 0) in Ω the solution (u , v ) of (5.5) converges to the trivial solution (0, 0) as t → ∞ if a1  d1 λ(1) , a2 < d2 λ(1) , and it converges to a semi-trivial solution in the form (0, v s ) if a  d1 λ(1) and a2 > d2 λ(1) , where λ(1) > 0 is the smallest eigenvalue of (5.4) corresponding to α = 0, β = m = 1. On the other hand, if a1 > d1 λ(1) and a2 < μ0 for a certain constant μ0 > 0 then the solution (u , v ) converges to another semi-trivial solution in the form (u s , 0) (cf. [25, p. 670]). This implies that under the above conditions on (a1 , a2 ) the trivial solution or one of the semi-trivial solutions is a global attractor of (5.5). However, the behavior of the solution is quite different for the degenerate case mi > 1, i = 1, 2. It is easy to see from Theorem 5.1 (with r = s = 1) that for any positive constants ai , b i and c i , Problem (5.6) has the trivial solution (0, 0) and two semi-trivial solutions (u s , 0) and (0, v s ), where u s and v s satisfy the respective relation 0 < u s (x) < a1 /b1 and 0 < v s (x) < a2 /c 2 . We show that under a simple condition on ai , b i and c i , independent of di and λ(1) , the above steady-state solutions are all unstable. In fact, we show that a unique global solution (u , v ) to (5.5) exists and enters the ¯ s  between a pair of quasisolutions us ≡ (u s , v s ) and u¯ s ≡ (u¯ s , v¯ s ) as t → ∞. sector us , u To show the above result we first construct a pair of coupled upper and lower solutions (u˜ s , v˜ s ), (uˆ s , vˆ s ) of (5.6). The requirement of these functions for the present problem is given by 1 ˜ s (a1 − b1 u˜ s − c 1 vˆ s ) −d1 u˜ m s u 2 ˜ s (a2 + b2 u˜ s − c 2 v˜ s ) −d2 v˜ m s v 1 ˆ s (a1 − b1 uˆ s − c 1 v˜ s ) −d1 uˆ m s u

(x ∈ Ω),

2 ˆ s (a2 + b2 uˆ s − c 2 vˆ s ) −d2 vˆ m s v

u˜ s (x)  0  uˆ s (x),

v˜ s (x)  0  vˆ s (x)

(x ∈ ∂Ω).

(5.7)

We seek such a pair in the form

(u˜ s , v˜ s ) = (ρ1 , ρ2 ),

  (uˆ s , vˆ s ) = (δφ0 )1/m1 , (δφ0 )1/m2 ,

(5.8)

where ρ1 , ρ2 and δ are some positive constants with δ sufficiently small and φ0 is the (normalized) positive eigenfunction of (5.4) (with α = 0, mβ = 1) corresponding to the smallest eigenvalue 1 2 ˆm λ0 > 0. Indeed, since (uˆ m s ,v s ) = (δφ0 , δφ0 ) and −(δφ0 ) = λ0 (δφ0 ), the pair in (5.8) satisfy all the inequalities in (5.7) if





0  ρ1 a1 − b1 ρ1 − c 1 (δφ0 )1/m2 , 0  ρ2 (a2 + b2 ρ1 − c 2 ρ2 ),





d1 λ0 (δφ0 )  (δφ0 )1/m1 a1 − b1 (δφ0 )1/m1 − c 1 ρ2 ,





d2 λ0 (δφ0 )  (δφ0 )1/m2 a2 + b2 (δφ0 )1/m1 − c 2 (δφ0 )1/m2 . It is obvious from the hypothesis m1 > 1, m2 > 1 that the above inequalities are all satisfied by a sufficiently small δ > 0 if

ρ1  a1 /b1 ,

ρ2  (a2 + b2 ρ1 )/c2 ,

ρ2 < a1 /c1 .

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

By choosing satisfying

1547

ρ1 = a1 /b1 , the above requirements are fulfilled by ρ2 = ρ ∗ where ρ ∗ is any constant a2 /c 2 + (b2 /b1 )(a1 /c 2 )  ρ ∗ < a1 /c 1 .

The existence of such a constant is ensured if a2 /a1 + b2 /b1 < c 2 /c 1 . Under this condition the pair in (5.8) with ρ1 = a1 /c 1 , ρ2 = ρ ∗ and a small constant δ > 0 are coupled upper and lower solutions. By Theorem 3.3, Problem (5.6) has a pair of quasisolutions (u¯ s , v¯ s ), (u s , v s ) that satisfy the equations 1 ¯ s (a1 − b1 u¯ s − c 1 v s ) −d1 u¯ m s =u 2 ¯ s (a2 + b2 u¯ s − c 2 v¯ s ) −d2  v¯ m s =v 1 ¯ s) −d1 um s = u s (a1 − b 1 u s − c 1 v

(x ∈ Ω),

2 −d2  v m s = v s (a2 + b 2 u s − c 2 v s )

u¯ s = u s = 0,

v¯ s = v s = 0

(x ∈ ∂Ω).

(5.9)

Since δ > 0 can be arbitrarily small and ρ ∗ arbitrarily close to a1 /c 1 an application of Theorem 3.3 and Theorem 4.3 leads to the following results. Theorem 5.2. Let ai , b i , c i , di and mi , i = 1, 2, be positive constants with mi > 1, and assume that

a2 /a1 + b2 /b1 < c 2 /c 1 .

(5.10)

Then the following statements hold true: (i) Problem (5.6) has a pair of quasisolutions (u¯ s , v¯ s ), (u s , v s ) that satisfy (5.9) and the relation

(0, 0) < (u s , v s )  (u¯ s , v¯ s ) < (a1 /b1 , a2 /c 1 ) in Ω. (ii) For any initial function (ψ1 , ψ2 ) satisfying

(0, 0) < (ψ1 , ψ2 ) < (a1 /b1 , a2 /c 1 ) in Ω,

(ψ1 , ψ2 ) = (0, 0) on ∂Ω

a unique global solution (u , v ) to (5.5) exists and possesses the property













u s (x), v s (x)  u (t , x), v (t , x)  u¯ s (x), v¯ s (x)

as t → ∞.

(iii) If (u¯ s , v¯ s ) = (u s , v s ) (≡ (u ∗s (x), v ∗s (x))) then (u ∗s , v ∗s ) is the unique solution of (5.6) in S ∗ and









lim u (t , x), v (t , x) = u ∗s (x), v ∗s (x)

t →∞

(x ∈ Ω).

Remark 5.1. (a) The conclusions in Theorem 5.2 imply that under the simple condition (5.10) and mi > 1, the predator and prey species coexist and enter the sector between a pair of quasisolutions as t → ∞. Moreover, for any constants ai (i = 1, 2), independent of the diffusion coefficients di and the domain Ω , the trivial solution (0, 0) and the semi-trivial solutions (u s , 0), (0, v s ) are unstable. This is in sharp contrast to the semilinear case m1 = m2 = 1 where one of these solutions is a global attractor when a1 or a2 is suitably small.

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(b) It is easy to see from the construction of upper and lower solutions for (5.6) that the conclusions in Theorem 5.2 hold true if one or both of the Dirichlet boundary conditions are replaced by the Neumann–Robin boundary conditions

∂ u /∂ ν + β1 u = 0 and ∂ v /∂ ν + β2 v = 0. In this situation the pair (u˜ s , v˜ s ) and (uˆ s , vˆ s ) in (5.8) remain to be coupled upper and lower solutions of (5.6) if φ0 is replaced by the strictly positive eigenfunction φ1 (φ1 and φ2 ) corresponding to the above boundary condition (or both conditions). Notice in this case that the condition (c) in (H3 ) is also satisfied. 5.3. A two-prey one-predator model Our final application is the following two-prey one-predator model with Neumann boundary condition:

∂ u /∂ t − d1 um1 = a1 u (1 − u − b1 v − c 1 w ) ∂ v /∂ t − d2  v m2 = a2 v (1 − v − b2 u − c 2 w ) ∂ w /∂ t − d3  w

m3

(t > 0, x ∈ Ω),

= a3 w (1 − w + b3 u + c 3 v )

∂ u /∂ ν = ∂ v /∂ ν = ∂ w /∂ ν = 0 u (0, x) = ψ1 (x),

(t > 0, x ∈ ∂Ω),

v (0, x) = ψ2 (x),

w (0, x) = ψ3 (x)

(x ∈ Ω)

(5.11)

where ai , b i , c i , di and mi , i = 1, 2, 3, are positive constants with mi  1. The above problem is a special case of (1.1) with N = 3, n0 = 1, u = (u 1 , u 2 , u 3 ) = (u , v , w ), and m i −1

D i (u i ) = m i u i

,

g i (x, u) = 0,

i = 1, 2, 3,

f 1 (u) = a1 u (1 − u − b1 v − c 1 w ), f 2 (u) = a2 v (1 − v − b2 u − c 2 w ), f 3 (u) = a3 w (1 − w + b3 u + c 3 v ).

(5.12)

The corresponding steady-state problem is given by

−d1 um1 = a1 u (1 − u − b1 v − c 1 w ) −d2  v m2 = a2 v (1 − v − b2 u − c 2 w ) −d3  w

m3

(x ∈ Ω),

= a3 w (1 − w + b3 u + c 3 v )

∂ u /∂ ν = ∂ v /∂ ν = ∂ w /∂ ν = 0

(x ∈ ∂Ω).

(5.13)

It is obvious that Problem (5.13) has the trivial solution (0, 0, 0) and various forms of semi-trivial solutions, including the constant solutions (1, 0, 0), (0, 1, 0) and (0, 0, 1). Since the function f(u) = ( f 1 (u), f 2 (u), f 3 (u)) in (5.12) is mixed quasimonotone for all u  0, the requirement of upper and ˜ s ≡ (u˜ s , v˜ s , w ˜ s ), uˆ s ≡ (uˆ s , vˆ s , w ˆ s ) becomes u˜ s  uˆ s > 0 in Ω and lower solutions u 1 ˜ s (1 − u˜ s − b1 vˆ s − c 1 w ˆ s ), −d1 u˜ m s  a1 u 2 ˜ s (1 − v˜ s − b2 uˆ s − c 2 w ˆ s ), −d2  v˜ m s  a2 v

m

˜ s (1 − w ˜ s 3  a3 w ˜ s + b3 u˜ s + c 3 v˜ s ), −d3  w

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

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1 ˆ s (1 − uˆ s − b1 v˜ s − c 1 w ˜ s ), −d1 uˆ m s  a1 u 2 ˆ s (1 − vˆ s − b2 u˜ s − c 2 w ˜ s ), −d2  vˆ m s  a2 v

m

ˆ s (1 − w ˆ s 3  a3 w ˆ s + b3 uˆ s + c 3 vˆ s ), −d3  w ∂ u˜ s /∂ ν  0  ∂ uˆ s /∂ ν ,

∂ v˜ s /∂ ν  0  ∂ vˆ s /∂ ν ,

˜ s /∂ ν  0  ∂ w ˆ s /∂ ν . ∂w

(5.14)

We construct a constant pair of coupled upper and lower solutions in the form

˜ s ) = ( M 1 , M 2 , M 3 ), (u˜ s , v˜ s , w

ˆ s ) = (δ1 , δ2 , δ3 ), (uˆ s , vˆ s , w

(5.15)

where M i and δi , i = 1, 2, 3, are positive constants to be chosen. Indeed, the above pair satisfy all the inequalities in (5.14) if

a1 M 1 (1 − M 1 − b1 δ2 − c 1 δ3 )  0  a1 δ1 (1 − δ1 − b1 M 2 − c 1 M 3 ), a2 M 2 (1 − M 2 − b2 δ1 − c 2 δ3 )  0  a2 δ2 (1 − δ2 − b2 M 1 − c 2 M 3 ), a3 M 3 (1 − M 3 + b3 M 1 + c 3 M 2 )  0  a3 δ3 (1 − δ3 + b3 δ1 + c 3 δ2 ). Choose M 1 = M 2 = 1 and δi (i = 1, 2, 3) sufficiently small. Then the first two inequalities at the lefthand side and the third inequality at the right-hand side are trivially satisfied, while the remaining inequalities are also satisfied if

b 1 + c 1 M 3 < 1,

b 2 + c 2 M 3 < 1,

1 + b3 + c 3  M 3 .

Assume that

 b 1 < 1,

b2 < 1 and

1 + b3 + c 3 < min

1 − b1 1 − b2

,

c1

c2

 .

(5.16)

Then the above requirement is fulfilled by any constant M 3 = M ∗ that satisfies

1 + b3 + c 3  M ∗ < min



1 − b1 1 − b2 c1

,

c2

 .

(5.17)

With this choice of M ∗ and some sufficiently small δi the constant pair (1, 1, M ∗ ) and (δ1 , δ2 , δ3 ) are coupled upper and lower solutions. By Theorem 3.3, the steady-state problem (5.13) has a pair of ¯ s ≡ (u¯ s , v¯ s , w ¯ s ), us ≡ (u s , v s , w s ) such that quasisolutions u

  ¯ s )  1, 1, M ∗ . (δ1 , δ2 , δ3 )  (u s , v s , w s )  (u¯ s , v¯ s , w

(5.18)

We show that if the matrix

 ∗

A ≡

−1 b2 b3

b1 −1 c3

c1 c2 −1

 (5.19)

¯ s = us (≡ ρ ∗ ) is a positive constant and u∗s ≡ ρ ∗ is the unique positive solution is nonsingular then u of (5.13). ¯ s = us ≡ ρ ∗ we observe from (5.12) and (u 1 , u 2 , u 3 ) = (u , v , w ) that the sequence of To prove u iterations in (3.6) for the present problem is given by

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C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

   (1 )  = c 1 I 1 u¯ (m−1) + a1 u¯ (m−1) 1 − u¯ (m−1) − b1 v (m−1) − c 1 w (m−1) ,     ¯ (2m) = c 2(1) I 2 v¯ (m−1) + a2 v¯ (m−1) 1 − v¯ (m−1) − b2 u (m−1) − c 2 w (m−1) , −L2 w  (m−1)    ¯ ¯ (3m) = c 3(1) I 3 w ¯ (m−1) 1 − w ¯ (m−1) + b3 u¯ (m−1) + c 3 v¯ (m−1) , + a3 w −L3 w     ¯ (m−1) , − L 1 w (1m) = c 1(1) I 1 u (m−1) + a1 u (m−1) 1 − u (m−1) − b1 v¯ (m−1) − c 1 w     ¯ (m−1) , − L 2 w (2m) = c 2(1) I 2 v (m−1) + a2 v (m−1) 1 − v (m−1) − b2 u¯ (m−1) − c 2 w     − L 3 w (3m) = c 3(1) I 3 w (m−1) + a3 w (m−1) 1 − w (m−1) + b3 u (m−1) + c 3 v (m−1) , (m)

¯1 −L1 w

¯ (i m) /∂ ν = ∂ w (i m) /∂ ν = 0, i = 1, 2, 3, ∂w  1/m1 (m)  1/m2 (m) ¯1 ¯2 u¯ (m) = w , v¯ (m) = w ,     ( m ) ( m ) 1 / m 1 / m u (m) = w 1 1 , v (m) = w 2 2 ,



1/m3 (m)

¯ (m) = w ¯3 w w (m) =



,

1/m (m) w3 3 ,

(5.20)

(1)

where c i , i = 1, 2, 3, are chosen as positive constants. Since the coupled initial iterations in the above ¯ (0) ) = (1, 1, M ∗ ) and (u (0) , v (0) , w (0) ) = (δ1 , δ2 , δ3 ), the iteration process are the constants (u¯ (0) , v¯ (0) , w (m)

¯i right-hand sides of (5.20) for m = 1 are all constants. In view of L i w i given by (2.8) for w i = w (m)

wi = wi

(1)

or

(1)

¯ i , u¯ i ) and , the uniqueness of the iterations in (5.20) ensures that the first iterations ( w

¯ (i m) , u¯ (i m) } and ( w (i 1) , u (i 1) ) are constants. In fact, an induction argument shows that the sequences { w (m)

{wi

(m)

, ui

} are constants and are given by

    = c 1(1) I 1 u¯ (m−1) + a1 u¯ (m−1) 1 − u¯ (m−1) − b1 v (m−1) − c 1 w (m−1) ,     (1) (m) ¯ 2 = c 2(1) I 2 v¯ (m−1) + a2 v¯ (m−1) 1 − v¯ (m−1) − b2 u (m−1) − c 2 w (m−1) , c2 w  (m−1)    (1) (m) ¯ ¯ 3 = c 3(1) I 3 w ¯ (m−1) 1 − w ¯ (m−1) + b3 u¯ (m−1) + c 3 v¯ (m−1) , + a3 w c3 w    (1) (m) (1 )  ¯ (m−1) , c 1 w 1 = c 1 I 1 u (m−1) + a1 u (m−1) 1 − u (m−1) − b1 v¯ (m−1) − c 1 w    (1) (m) (1 )  ¯ (m−1) , c 2 w 2 = c 2 I 2 v (m−1) + a2 v (m−1) 1 − v (m−1) − b2 u¯ (m−1) − c 2 w    (1) (m) (1 )  c 3 w 3 = c 3 I 3 w (m−1) + a3 w (m−1) 1 − w (m−1) + b3 u (m−1) + c 3 v (m−1) . (1 )

(m)

¯1 c1 w

(5.21)

Notice from the last two relations in (5.20) that

  = I 1 u¯ (m) ,   (m) w 1 = I 1 u (m) ,

  = I 2 v¯ (m) ,   (m) w 2 = I 2 v (m) ,

(m)

(m)

¯i Since by Theorem 3.1, the sequences { w (m)

(m)

 (m)  ¯ , = I3 w   (m) w 3 = I 3 w (m) .

(m)

(m)

¯2 w

¯1 w

(m)

, u¯ i

(m)

(m)

}, { w i

¯3 w

(m)

, ui

(5.22)

} governed by (5.21) converge monoton-

ically to some limits where (u 1 , u 2 , u 3 ) = (u (m) , v (m) , w (m) ) we see that the limits are constants. In fact,





¯ (m) = (ρ¯1 , ρ¯2 , ρ¯3 ), lim u¯ (m) , v¯ (m) , w

m→∞

for some constants ρ¯i ,





lim u (m) , v (m) , w (m) = (ρ 1 , ρ 2 , ρ 3 )

m→∞

ρ i , i = 1, 2, 3, satisfying the relation   (δ1 , δ2 , δ3 )  (ρ 1 , ρ 2 , ρ 3 )  (ρ¯1 , ρ¯2 , ρ¯3 )  1, 1, M ∗ .

(5.23)

C.V. Pao, W.H. Ruan / J. Differential Equations 255 (2013) 1515–1553

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Letting m → ∞ in (5.21) and using the relation (5.22) shows that the limits in (5.23) satisfy the equations

1 − ρ¯1 − b1 ρ 2 − c 1 ρ 3 = 0,

1 − ρ 1 − b1 ρ¯2 − c 1 ρ¯3 = 0,

1 − ρ¯2 − b2 ρ 1 − c 2 ρ 3 = 0,

1 − ρ 2 − b2 ρ¯1 − c 2 ρ¯3 = 0,

1 − ρ¯3 + b3 ρ¯1 + c 3 ρ¯2 = 0,

1 − ρ 3 + b 3 ρ 1 + c 3 ρ 2 = 0.

Let ρi = ρ¯i − ρ i , i = 1, 2, 3. Then a subtraction of the corresponding equations in the above relation leads to

−ρ1 + b1 ρ2 + c 1 ρ3 = 0,

b2 ρ1 − ρ2 + c 2 ρ3 = 0,

b3 ρ1 + c 3 ρ2 − ρ3 = 0.

It follows from the nonsingular property of the matrix A ∗ in (5.19) that ρ1 = ρ2 = ρ3 = 0. This shows that (ρ¯1 , ρ¯2 , ρ¯3 ) = (ρ 1 , ρ 2 , ρ 3 ) ≡ (ρ1∗ , ρ2∗ , ρ3∗ ) and ρ ∗ ≡ (ρ1∗ , ρ2∗ , ρ3∗ ) is the unique constant solution of Problem (5.13) in S ∗ . By an application of Theorem 3.3 and Theorem 4.3 we have the following

Theorem 5.3. Let ai , b i , c i , di and mi , i = 1, 2, 3, be positive constants with mi  1, and assume that the matrix A ∗ in (5.19) is nonsingular and condition (5.16) holds. Then the following statements hold true: (i) The steady-state problem (5.13) has a unique positive constant solution (u ∗s , v ∗s , w ∗s ) ≡ (ρ1∗ , ρ2∗ , ρ3∗ )  (1, 1, M ∗ ), where M ∗ satisfies (5.17). (ii) Problem (5.13) has no non-constant positive solution in the sector δ, M∗  where δ ≡ (δ1 , δ2 , δ3 ) and M∗ ≡ (1, 1, M ∗ ). (iii) For any initial function ψ ≡ (ψ1 , ψ2 , ψ3 ) in δ, M∗  the time-dependent problem (5.11) has a unique global solution (u (t , x), v (t , x), w (t , x)). Moreover





lim u (t , x), v (t , x), w (t , x) =

t →∞



ρ1∗ , ρ2∗ , ρ3∗



(x ∈ Ω).

Remark 5.2. Theorem 5.3 implies that the constant steady-state solution ρ ∗ ≡ (ρ1∗ , ρ2∗ , ρ3∗ ) is a global attractor (relative to δ, M∗ ). Since δ can be chosen arbitrarily small, the trivial solution (0, 0, 0) and all forms of semi-trivial solutions, including the constant semi-trivial solutions (1, 0, 0), (0, 1, 0) and (0, 0, 1), are all unstable. It is obvious that the nonsingular property of A ∗ and condition (5.16) are satisfied when b i and c i are small (relative to one). This implies that for small interaction rates in the reaction function the population species u, v and w coexist and converge to a unique positive constant equilibrium as t → ∞. This convergence property holds true for any positive constants ai , di and mi  1, including the case mi = 1 for some or all i. It is also true for any size of the domain Ω . References [1] H. Amann, Dynamic theory of quasilinear parabolic systems II. Reaction–diffusion systems, Differential Integral Equations 3 (1990) 13–75. [2] J.R. Anderson, Local existence and uniqueness of solutions of degenerate parabolic equations, Comm. Partial Differential Equations 16 (1991) 105–143. [3] J.R. Anderson, K. Deng, Global existence for nonlinear diffusion equations, J. Math. Anal. Appl. 196 (1995) 479–501. [4] B. Andreianov, M. Bendahmane, K.H. Karlsen, S. Ouaro, Well-posedness results for triply nonlinear degenerate parabolic equations, J. Differential Equations 247 (2009) 277–302. [5] D.G. Aronson, L.A. Peletier, Large time behavior of solutions of the porous medium equation in bounded domain, J. Differential Equations 39 (1981) 378–412. [6] G.I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Mekh. 16 (1952) 67–78 (in Russian). [7] R. Bürger, H. Frid, K.H. Karlsen, On a free boundary problem for a strongly degenerate quasi-linear parabolic equation with an application to a model of pressure filtration, SIAM J. Math. Anal. 34 (3) (2003) 611–635.

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