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A comparison principle for nonlocal coupled systems of fully nonlinear parabolic equations
Applied Mathematics Letters 19 (2006) 1272–1277 www.elsevier.com/locate/aml
A comparison principle for nonlocal coupled systems of fully nonlinear parabolic equations✩ Rong-Nian Wang ∗ , Ti-Jun Xiao, Jin Liang Department of Mathematics, University of Science and Technology of China, Hefei 230026, People’s Republic of China Received 14 March 2005; received in revised form 1 January 2006; accepted 25 January 2006
Abstract In this note, we establish a quite general comparison principle for a class of coupled systems of fully nonlinear parabolic equations subject to nonlocal boundary conditions. c 2006 Elsevier Ltd. All rights reserved. Keywords: Comparison principle; Nonlocal boundary conditions; Coupled system; Fully nonlinear parabolic equations
1. Introduction Let Ω ⊂ Rn , n ≥ 1, be a bounded domain with the boundary ∂Ω . Set D M = (0, M) × Ω ,
SM = (0, M) × ∂Ω ,
where M > 0 is an arbitrary constant. Of concern is the following nonlocal coupled system of fully nonlinear parabolic equations: ∂u i = f i (t, x, u, ∇u i , ∇ 2 u i ) in D M , ∂t N Bi u i = K i j (x, y)u j (t, y)dy on SM ,
(1.1)
Ω j =1
u i (t, x) = u i0 (x)
in Ω , (i = 1, . . . , N),
where u = (u 1 , . . . , u N ), ∇u i and ∇ 2 u i denote, respectively, the gradient of u i and the Hessian matrix of u i in space 2 variables, and for each i = 1, . . . , N, fi ∈ C[D M × R N × Rn × Rn , R], Bi is given by Bi = αi (x)
∂ + 1, ∂ν
(1.2)
✩ This work was supported partly by the National NSF of China, the Key Project Foundation of the Chinese Academy of Sciences, and the Ministry of Education of China. ∗ Corresponding author. E-mail addresses: [email protected] (R.-N. Wang), [email protected] (T.-J. Xiao), [email protected] (J. Liang).
c 2006 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2006.01.012
R.-N. Wang et al. / Applied Mathematics Letters 19 (2006) 1272–1277
1273
∂ ∂ν
being the outward normal derivative on ∂Ω . Suppose that for each i = 1, . . . , N, f i is an elliptic operator in D M (see Section 2 below for its definition), Bi is either of Dirichlet type (αi = 0) or of Robin type (αi > 0), and it is allowed to be of different types for different i . It is well known that various comparison principles play a crucial role for studying properties of equations of parabolic type (cf. [1–13] and references therein). In [13], for the case N = 1, Yin established a certain comparison principle for system (1.1) under the boundary condition u(t, x) = Ω K (x, y)u(t, y)dy (t > 0, x ∈ ∂Ω ). The comparison principle was studied by Pao in [7,9] for a scalar semilinear parabolic equation with the following boundary condition: ∂u α(x) (t, x) + u(t, x) = K (x, y)u(t, y)dy (t > 0, x ∈ ∂Ω ), ∂ν Ω where α(x) ≥ 0, and in [8] for the system of a finite number of semilinear parabolic equations with the following boundary conditions: Bi u i = K i (x, y)u i (t, y)dy + h i (x) (t > 0, x ∈ ∂Ω ), i = 1, . . . , N. Ω
Motivated by the above works, in this note, we establish a general comparison principle for the more complex coupled system (1.1), which is an extension of the corresponding results in [7–9,13]. 2. Results and proofs We set f(t, x, u) ≡ ( f 1 (t, x, u, ∇u 1 , ∇ 2 u 1 ), . . . , f N (t, x, u, ∇u N , ∇ 2 u N )), S11 · · · S1n T11 · · · T1n .. , .. . S = ... T = ... . . Sn1
···
Snn
Tn1
···
Tnn
Definition 2.1. A vector function f(t, x, u) is said to be elliptic at a point (t1 , x 1 ) if for each i = 1, . . . , N, the function 2 fi ∈ C[D M × R N × Rn × Rn , R] is elliptic at (t1 , x 1 ), that is, for any u, P, S j k , T j k ( j, k = 1, . . . , n), the quadratic n
(S j k − T j k )λ j λk ≤ 0
for arbitrary vector λ ∈ Rn
j,k=1
implies f i (t1 , x 1 , u, P, S) ≤ f i (t1 , x 1 , u, P, T ). If the vector function f(t, x, u) is elliptic for every (t, x) ∈ D M , then f(t, x, u) is said to be elliptic in D M . Definition 2.2. A vector function f(t, x, u) is said to be quasi-monotone nondecreasing in some subset J of R N if for each i = 1, . . . , N, on writing u in the split form u = (u i , [u] N−1 ), fi (t, x, u i , [u] N−1 , ∇u i , ∇ 2 u i ) is obtained as nondecreasing with respect to the components of [u] N−1 for all u ∈ J . Throughout this work we assume: (H1 ) The vector function f(t, x, u) is elliptic in D M and is quasi-monotone nondecreasing in some given subset of RN . (H2 ) For each i, j = 1, . . . , N, N K i j (x, y) > 0, K i j (x, y)dy ≤ 1 (x ∈ ∂Ω , y ∈ Ω ). Ω j =1
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R.-N. Wang et al. / Applied Mathematics Letters 19 (2006) 1272–1277
(H3) For each i = 1, . . . , N, fi satisfies the one-side Lipschitz condition f i (t, x, u, P, S) − fi (t, x, v, P, S) ≤ Ni
N
(u i − vi ),
if u ≥ v,
j =1
where u, v ∈ R N and Ni > 0 is a constant. Denote by C 1,2 (D M ) the set of functions that are once continuously differentiable in t and twice continuously differentiable in x for (t, x) ∈ D M and by C 0,1 (D M ) the set of functions that are continuous in t and once continuously differentiable in x for (t, x) ∈ D M . Now we are in a position to show our main result. Theorem 2.1. Let the hypotheses (H1 ), (H2) and (H3 ) hold, and let u, v ∈ (C 1,2 (D M ) ∩ C 0,1 (D M )) N satisfy u it ≤ f i (t, x, u, ∇u i , ∇ 2 u i )
in D M ,
vit ≥ f i (t, x, v, ∇vi , ∇ vi ) in D M , N Bi u i ≤ K i j (x, y)u j (t, y)dy + h i (t, x) on SM , 2
Ω j =1
Bi vi ≥
N Ω j =1
(2.1)
K i j (x, y)v j (t, y)dy + h i (t, x) on SM ,
u i (0, x) ≤ vi (0, x) in Ω , (i = 1, . . . , N), where u = (u 1 , . . . , u N ) and v = (v1 , . . . , v N ). It then follows that u(t, x) ≤ v(t, x) in D M . Proof. Step 1. Write w(t, x) := (w1 (t, x), . . . , w N (t, x)), where for each i = 1, . . . , N, wi (t, x) = u i (t, x) − vi (t, x) and u = (u 1 , . . . , u N ) and v = (v1 , . . . , v N ) satisfy the following inequalities: u it ≤ f i (t, x, u, ∇u i , ∇ 2 u i )
in D M ,
vit > f i (t, x, v, ∇vi , ∇ vi ) in D M , N Bi u i ≤ K i j (x, y)u j (t, y)dy + h i (t, x) on SM , 2
Ω j =1
Bi vi ≥
N Ω j =1
(2.2)
K i j (x, y)v j (t, y)dy + h i (t, x) on SM ,
u i (0, x) < vi (0, x) in Ω , (i = 1, . . . , N). We prove w(t, x) < 0
on D M .
(2.3)
Since w(0, x) = u(0, x)−v(0, x) < 0 (x ∈ Ω ), by continuity, there exists a δ > 0 such that w(t, x) < 0 for 0 ≤ t ≤ δ and x ∈ Ω . Let Γ = {t; t ≤ M, w(s, x) < 0 for 0 ≤ s ≤ t and x ∈ Ω}, then there exists t1 = sup Γ and 0 < t1 ≤ M. Hence, w(t, x) ≤ 0 on D t1 . If (2.3) is not true, then there exists a point x 1 ∈ Ω and wi (t, x) such that wi (t, x) < 0,
on [0, t1 )
and wi (t1 , x 1 ) = 0.
That is, t1 (> 0) is the first time at which wi (t, x) has a zero for some x 1 ∈ Ω . Therefore, wi (t, x) attains its maximum at (t1 , x 1 ) on D t1 . We first show (t1 , x 1 ) ∈ D M .
R.-N. Wang et al. / Applied Mathematics Letters 19 (2006) 1272–1277
If (t1 , x 1 ) ∈ D M , then
∂wi ∂t (t1 , x 1 )
n ∂ 2 wi (t1 , x 1 )λi λ j ≤ 0, ∂ xi x j i, j =1
≥ 0,
∂wi ∂ xi
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(t1 , x 1 ) = 0, and
∀λ = (λ1 , . . . , λ N ) ∈ Rn .
Using the quasi-monotone nondecreasing property and the ellipticity of f(t, x, u), (2.2) implies 0≤
∂wi (t1 , x 1 ) < f i (t1 , x 1 , u i (t1 , x 1 ), [u] N−1 (t1 , x 1 ), ∇u i (t1 , x 1 ), ∇ 2 u i (t1 , x 1 )) ∂t − f i (t1 , x 1 , vi (t1 , x 1 ), [v] N−1 (t1 , x 1 ), ∇vi (t1 , x 1 ), ∇ 2 vi (t1 , x 1 )) ≤ f i (t1 , x 1 , u i (t1 , x 1 ), [v] N−1 (t1 , x 1 ), ∇u i (t1 , x 1 ), ∇ 2 u i (t1 , x 1 )) − f i (t1 , x 1 , vi (t1 , x 1 ), [v] N−1 (t1 , x 1 ), ∇vi (t1 , x 1 ), ∇ 2 vi (t1 , x 1 )) ≤ 0,
which is a contradiction. Therefore, w(t, x) < 0 on D M . Also, (t1 , x 1 ) ∈ SM ; otherwise, for the case αi = 0 (Dirichlet boundary condition), 0 = wi (t1 , x 1 ) = u i (t1 , x 1 ) − vi (t1 , x 1 ) ≤
N Ω j =1
K i j (x 1 , y)(u j (t1 , y) − v j (t1 , y))dy < 0,
which leads to a contradiction; at the same time, it follows from wi (t1 , x 1 ) = max D wi (t, x), wi (t, x) < 0 = t wi (t1 , x 1 ) in Dt , that ∂wi (t1 , x 1 ) ≥ 0. ∂ν Hence, for the case αi > 0 (the Robin boundary condition), 0 ≤ αi
N ∂wi (t1 , x 1 ) + wi (t1 , x 1 ) ≤ K i j (x 1 , y)(u j (t1 , y) − v j (t1 , y))dy < 0, ∂ν Ω j =1
which leads to a contradiction again. This proves w(t, x) < 0 on D M . Step 2. For the general case where (2.1) holds, let z(t) = (eγ c1 N N1 t , . . . , eγ c N N N N t ). Here, we choose positive constants γ , ci (i = 1, . . . , N) such that c1 N1 = c2 N2 = · · · = c N N N and for each i = 1, . . . , N, γ ci > 1. Consider v = v + εz for any small ε > 0. By the one-side Lipschitz condition, we have vit = vit + εz it ≥ f i (t, x, v, ∇vi , ∇ 2 vi ) + γ εci N Ni eγ ci N Ni t ≥ f i (t, x, v , ∇vi , ∇ 2 vi ) − ε Ni
N
eγ c j N N j t + γ εci N Ni eγ ci N Ni t
j =1
= f i (t, x, v , ∇vi , ∇ vi ) − ε N Ni eγ ci N Ni t + εγ ci N Ni e2ci N Ni t > f i (t, x, v , ∇vi , ∇ 2 vi ) 2
= f i (t, x, v , ∇vi , ∇ 2 vi ) and for each i = 1, . . . , N, u i (0, x) < vi (0, x),
∀ x ∈ Ω.
in D M , (i = 1, . . . , N),
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R.-N. Wang et al. / Applied Mathematics Letters 19 (2006) 1272–1277
Also, by the hypothesis (H2 ), Bi vi = αi
N ∂vi (t, x) + v (t, x) ≥ K i j (x, y)v j (t, y)dy + εeγ ci N Ni t + h i (t, x) i ∂vi Ω j =1 N ≥ K i j (x, y)v j (t, y)dy + h i (t, x). Ω j =1
v
That is, u and satisfy the inequalities in (2.2), with v replaced by v . Therefore, by Step 1, u(t, x) < v (t, x) on D M . Taking ε → 0 yields the desired result and this completes the proof. Remark 2.1. Theorem 2.1 extends the corresponding results in [7–9] and [13]. The upper and lower solutions to system (1.1) are defined as follows. Definition 2.3. A vector function
u = (
u 1 , . . . ,
u N ) ∈ C 1,2(D M ) ∩ C(D M ) is called an upper solution of system (1.1) u satisfies on D M if
∂
ui u i ) in D M , ≥ fi (t, x,
u, ∇
u i , ∇ 2
∂t N Bi
K i j (x, y)
u j (t, y)dy on Γ M , ui ≥ Ω j =1
u i (t, x) ≥ u i0 (x)
in Ω , (i = 1, . . . , N).
A lower solution is defined analogously by reversing the above inequalities. As an immediate consequence of Theorem 2.1, we have: Corollary 2.1. Let the hypotheses (H1), (H2) and (H3) hold. Suppose that
u, u ∈ (C 1,2 (D M ) ∩ C 0,1 (D M )) N are upper and lower solutions of system (1.1) and u is any solution of system (1.1), then, u ≤ u ≤
u on D M . Corollary 2.1 concerns the order-preserving property of the upper and lower solutions. Moreover, it is easy to get the following corollary on the uniqueness of a solution to system (1.1). Corollary 2.2. Suppose system (1.1) has a solution and the hypotheses (H1), (H2 ) and (H3) hold. Then the solution for system (1.1) is unique. Example 2.1. Consider the following coupled system of convection–reaction–diffusion equations: ∂u i − L i (t)u i = gi (t, x, u, ∇u i ) (t > 0, x ∈ Ω ), ∂t N Bi u i = K i j (x, y)u j (t, y)dy (t > 0, x ∈ ∂Ω ),
(2.4)
Ω j =1
u i (t, x) = u i0 (x)
in Ω , (i = 1, . . . , N),
where the elliptic operator L i (t) is given by L i (t) =
n j,k=1
(i)
a j k (t, x)
n ∂2 ∂ (i) + b j (t, x) , ∂ x j ∂ xk ∂ xj j =1
(i = 1, . . . , N).
In view of Theorem 2.1, we obtain the following new information about the coupled system (2.4): Assume for each i, j = 1, . . . , N, the kernel K i j (x, y) satisfies the hypothesis (H2), function gi (·, u 1 , . . . , u N , ·) is nondecreasing with respect to u j ( j = i ) and satisfies the one-side Lipschitz condition gi (t, x, u, P) − gi (t, x, v, P) ≤ di
N (u i − vi ), j =1
if u ≥ v,
R.-N. Wang et al. / Applied Mathematics Letters 19 (2006) 1272–1277
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where u, v ∈ R N , P is any vector in Rn , and di > 0 is a constant. Suppose
u, u ∈ (C 1,2 (D M ) ∩ C 0,1 (D M )) N are upper and lower solutions of system (2.4) and u is any solution of system (2.4); then, u ≤ u ≤
u and u is a unique solution. Acknowledgments The authors thank Prof. C. V. Pao very much for sending us reprints of [8,9], which were helpful to us. Moreover, the authors are grateful to the anonymous referee for his/her careful reading and valuable suggestions. References [1] S. Carl, V. Lakshmikantham, Generalized quasilinearization method for reaction–diffusion equations under nonlinear and nonlocal flux conditions, J. Math. Anal. Appl. 271 (2002) 182–205. [2] W.A. Day, Extensions of a property of heat equation to linear thermoelasticity and other theories, Quart. Appl. Math. 40 (1982) 319–330. [3] W.A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math. 40 (1983) 468–475. [4] L.C. Evans, Partial differential equations, in: Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. [5] O.A. Ladyzenskaja, V. Solonnikov, N. Uralceva, Linear and quasilinear equations of parabolic type, Amer. Math. Soc. Transl. (1968). [6] D.G. Levitt, Interpretation of biological ion channel flux data: reaction rate versus continuum theory, Annu. Rev. Biophys. Chem. 15 (1986) 29–57. [7] C.V. Pao, Dynamics of reaction–diffusion equations with nonlocal boundary conditions, Quart. Appl. Math. 50 (1995) 173–186. [8] C.V. Pao, Dynamics of weakly coupled parabolic systems with nonlocal boundary conditions, in: Advances in Nonlinear Dynamics, in: Stability Control Theory Methods Appl., vol. 5, Gordon and Breach, Amsterdam, 1997, pp. 319–327. [9] C.V. Pao, Asymptotic behavior of solutions of reaction–diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math. 88 (1998) 225–238. [10] M.F. Schumaker, Boundary conditions and trajectories of diffusion process, J. Chem. Phys. 117 (2002) 2469–2473. [11] S. Seo, Global existence and decreasing property of boundary values of solutions to parabolic equations with nonlocal boundary conditions, Pacific. J. Math. 193 (2000) 219–226. [12] H.M. Yin, On a class of parabolic equations with nonlocal boundary conditions, J. Math. Anal. Appl. 294 (2004) 712–728. [13] Y.F. Yin, On nonlinear parabolic equations with nonlocal boundary condition, J. Math. Anal. Appl. 185 (1994) 54–60.
A comparison principle for nonlocal coupled systems of fully nonlinear parabolic equations✩ Rong-Nian Wang ∗ , Ti-Jun Xiao, Jin Liang Department of Mathematics, University of Science and Technology of China, Hefei 230026, People’s Republic of China Received 14 March 2005; received in revised form 1 January 2006; accepted 25 January 2006
Abstract In this note, we establish a quite general comparison principle for a class of coupled systems of fully nonlinear parabolic equations subject to nonlocal boundary conditions. c 2006 Elsevier Ltd. All rights reserved. Keywords: Comparison principle; Nonlocal boundary conditions; Coupled system; Fully nonlinear parabolic equations
1. Introduction Let Ω ⊂ Rn , n ≥ 1, be a bounded domain with the boundary ∂Ω . Set D M = (0, M) × Ω ,
SM = (0, M) × ∂Ω ,
where M > 0 is an arbitrary constant. Of concern is the following nonlocal coupled system of fully nonlinear parabolic equations: ∂u i = f i (t, x, u, ∇u i , ∇ 2 u i ) in D M , ∂t N Bi u i = K i j (x, y)u j (t, y)dy on SM ,
(1.1)
Ω j =1
u i (t, x) = u i0 (x)
in Ω , (i = 1, . . . , N),
where u = (u 1 , . . . , u N ), ∇u i and ∇ 2 u i denote, respectively, the gradient of u i and the Hessian matrix of u i in space 2 variables, and for each i = 1, . . . , N, fi ∈ C[D M × R N × Rn × Rn , R], Bi is given by Bi = αi (x)
∂ + 1, ∂ν
(1.2)
✩ This work was supported partly by the National NSF of China, the Key Project Foundation of the Chinese Academy of Sciences, and the Ministry of Education of China. ∗ Corresponding author. E-mail addresses: [email protected] (R.-N. Wang), [email protected] (T.-J. Xiao), [email protected] (J. Liang).
c 2006 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2006.01.012
R.-N. Wang et al. / Applied Mathematics Letters 19 (2006) 1272–1277
1273
∂ ∂ν
being the outward normal derivative on ∂Ω . Suppose that for each i = 1, . . . , N, f i is an elliptic operator in D M (see Section 2 below for its definition), Bi is either of Dirichlet type (αi = 0) or of Robin type (αi > 0), and it is allowed to be of different types for different i . It is well known that various comparison principles play a crucial role for studying properties of equations of parabolic type (cf. [1–13] and references therein). In [13], for the case N = 1, Yin established a certain comparison principle for system (1.1) under the boundary condition u(t, x) = Ω K (x, y)u(t, y)dy (t > 0, x ∈ ∂Ω ). The comparison principle was studied by Pao in [7,9] for a scalar semilinear parabolic equation with the following boundary condition: ∂u α(x) (t, x) + u(t, x) = K (x, y)u(t, y)dy (t > 0, x ∈ ∂Ω ), ∂ν Ω where α(x) ≥ 0, and in [8] for the system of a finite number of semilinear parabolic equations with the following boundary conditions: Bi u i = K i (x, y)u i (t, y)dy + h i (x) (t > 0, x ∈ ∂Ω ), i = 1, . . . , N. Ω
Motivated by the above works, in this note, we establish a general comparison principle for the more complex coupled system (1.1), which is an extension of the corresponding results in [7–9,13]. 2. Results and proofs We set f(t, x, u) ≡ ( f 1 (t, x, u, ∇u 1 , ∇ 2 u 1 ), . . . , f N (t, x, u, ∇u N , ∇ 2 u N )), S11 · · · S1n T11 · · · T1n .. , .. . S = ... T = ... . . Sn1
···
Snn
Tn1
···
Tnn
Definition 2.1. A vector function f(t, x, u) is said to be elliptic at a point (t1 , x 1 ) if for each i = 1, . . . , N, the function 2 fi ∈ C[D M × R N × Rn × Rn , R] is elliptic at (t1 , x 1 ), that is, for any u, P, S j k , T j k ( j, k = 1, . . . , n), the quadratic n
(S j k − T j k )λ j λk ≤ 0
for arbitrary vector λ ∈ Rn
j,k=1
implies f i (t1 , x 1 , u, P, S) ≤ f i (t1 , x 1 , u, P, T ). If the vector function f(t, x, u) is elliptic for every (t, x) ∈ D M , then f(t, x, u) is said to be elliptic in D M . Definition 2.2. A vector function f(t, x, u) is said to be quasi-monotone nondecreasing in some subset J of R N if for each i = 1, . . . , N, on writing u in the split form u = (u i , [u] N−1 ), fi (t, x, u i , [u] N−1 , ∇u i , ∇ 2 u i ) is obtained as nondecreasing with respect to the components of [u] N−1 for all u ∈ J . Throughout this work we assume: (H1 ) The vector function f(t, x, u) is elliptic in D M and is quasi-monotone nondecreasing in some given subset of RN . (H2 ) For each i, j = 1, . . . , N, N K i j (x, y) > 0, K i j (x, y)dy ≤ 1 (x ∈ ∂Ω , y ∈ Ω ). Ω j =1
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R.-N. Wang et al. / Applied Mathematics Letters 19 (2006) 1272–1277
(H3) For each i = 1, . . . , N, fi satisfies the one-side Lipschitz condition f i (t, x, u, P, S) − fi (t, x, v, P, S) ≤ Ni
N
(u i − vi ),
if u ≥ v,
j =1
where u, v ∈ R N and Ni > 0 is a constant. Denote by C 1,2 (D M ) the set of functions that are once continuously differentiable in t and twice continuously differentiable in x for (t, x) ∈ D M and by C 0,1 (D M ) the set of functions that are continuous in t and once continuously differentiable in x for (t, x) ∈ D M . Now we are in a position to show our main result. Theorem 2.1. Let the hypotheses (H1 ), (H2) and (H3 ) hold, and let u, v ∈ (C 1,2 (D M ) ∩ C 0,1 (D M )) N satisfy u it ≤ f i (t, x, u, ∇u i , ∇ 2 u i )
in D M ,
vit ≥ f i (t, x, v, ∇vi , ∇ vi ) in D M , N Bi u i ≤ K i j (x, y)u j (t, y)dy + h i (t, x) on SM , 2
Ω j =1
Bi vi ≥
N Ω j =1
(2.1)
K i j (x, y)v j (t, y)dy + h i (t, x) on SM ,
u i (0, x) ≤ vi (0, x) in Ω , (i = 1, . . . , N), where u = (u 1 , . . . , u N ) and v = (v1 , . . . , v N ). It then follows that u(t, x) ≤ v(t, x) in D M . Proof. Step 1. Write w(t, x) := (w1 (t, x), . . . , w N (t, x)), where for each i = 1, . . . , N, wi (t, x) = u i (t, x) − vi (t, x) and u = (u 1 , . . . , u N ) and v = (v1 , . . . , v N ) satisfy the following inequalities: u it ≤ f i (t, x, u, ∇u i , ∇ 2 u i )
in D M ,
vit > f i (t, x, v, ∇vi , ∇ vi ) in D M , N Bi u i ≤ K i j (x, y)u j (t, y)dy + h i (t, x) on SM , 2
Ω j =1
Bi vi ≥
N Ω j =1
(2.2)
K i j (x, y)v j (t, y)dy + h i (t, x) on SM ,
u i (0, x) < vi (0, x) in Ω , (i = 1, . . . , N). We prove w(t, x) < 0
on D M .
(2.3)
Since w(0, x) = u(0, x)−v(0, x) < 0 (x ∈ Ω ), by continuity, there exists a δ > 0 such that w(t, x) < 0 for 0 ≤ t ≤ δ and x ∈ Ω . Let Γ = {t; t ≤ M, w(s, x) < 0 for 0 ≤ s ≤ t and x ∈ Ω}, then there exists t1 = sup Γ and 0 < t1 ≤ M. Hence, w(t, x) ≤ 0 on D t1 . If (2.3) is not true, then there exists a point x 1 ∈ Ω and wi (t, x) such that wi (t, x) < 0,
on [0, t1 )
and wi (t1 , x 1 ) = 0.
That is, t1 (> 0) is the first time at which wi (t, x) has a zero for some x 1 ∈ Ω . Therefore, wi (t, x) attains its maximum at (t1 , x 1 ) on D t1 . We first show (t1 , x 1 ) ∈ D M .
R.-N. Wang et al. / Applied Mathematics Letters 19 (2006) 1272–1277
If (t1 , x 1 ) ∈ D M , then
∂wi ∂t (t1 , x 1 )
n ∂ 2 wi (t1 , x 1 )λi λ j ≤ 0, ∂ xi x j i, j =1
≥ 0,
∂wi ∂ xi
1275
(t1 , x 1 ) = 0, and
∀λ = (λ1 , . . . , λ N ) ∈ Rn .
Using the quasi-monotone nondecreasing property and the ellipticity of f(t, x, u), (2.2) implies 0≤
∂wi (t1 , x 1 ) < f i (t1 , x 1 , u i (t1 , x 1 ), [u] N−1 (t1 , x 1 ), ∇u i (t1 , x 1 ), ∇ 2 u i (t1 , x 1 )) ∂t − f i (t1 , x 1 , vi (t1 , x 1 ), [v] N−1 (t1 , x 1 ), ∇vi (t1 , x 1 ), ∇ 2 vi (t1 , x 1 )) ≤ f i (t1 , x 1 , u i (t1 , x 1 ), [v] N−1 (t1 , x 1 ), ∇u i (t1 , x 1 ), ∇ 2 u i (t1 , x 1 )) − f i (t1 , x 1 , vi (t1 , x 1 ), [v] N−1 (t1 , x 1 ), ∇vi (t1 , x 1 ), ∇ 2 vi (t1 , x 1 )) ≤ 0,
which is a contradiction. Therefore, w(t, x) < 0 on D M . Also, (t1 , x 1 ) ∈ SM ; otherwise, for the case αi = 0 (Dirichlet boundary condition), 0 = wi (t1 , x 1 ) = u i (t1 , x 1 ) − vi (t1 , x 1 ) ≤
N Ω j =1
K i j (x 1 , y)(u j (t1 , y) − v j (t1 , y))dy < 0,
which leads to a contradiction; at the same time, it follows from wi (t1 , x 1 ) = max D wi (t, x), wi (t, x) < 0 = t wi (t1 , x 1 ) in Dt , that ∂wi (t1 , x 1 ) ≥ 0. ∂ν Hence, for the case αi > 0 (the Robin boundary condition), 0 ≤ αi
N ∂wi (t1 , x 1 ) + wi (t1 , x 1 ) ≤ K i j (x 1 , y)(u j (t1 , y) − v j (t1 , y))dy < 0, ∂ν Ω j =1
which leads to a contradiction again. This proves w(t, x) < 0 on D M . Step 2. For the general case where (2.1) holds, let z(t) = (eγ c1 N N1 t , . . . , eγ c N N N N t ). Here, we choose positive constants γ , ci (i = 1, . . . , N) such that c1 N1 = c2 N2 = · · · = c N N N and for each i = 1, . . . , N, γ ci > 1. Consider v = v + εz for any small ε > 0. By the one-side Lipschitz condition, we have vit = vit + εz it ≥ f i (t, x, v, ∇vi , ∇ 2 vi ) + γ εci N Ni eγ ci N Ni t ≥ f i (t, x, v , ∇vi , ∇ 2 vi ) − ε Ni
N
eγ c j N N j t + γ εci N Ni eγ ci N Ni t
j =1
= f i (t, x, v , ∇vi , ∇ vi ) − ε N Ni eγ ci N Ni t + εγ ci N Ni e2ci N Ni t > f i (t, x, v , ∇vi , ∇ 2 vi ) 2
= f i (t, x, v , ∇vi , ∇ 2 vi ) and for each i = 1, . . . , N, u i (0, x) < vi (0, x),
∀ x ∈ Ω.
in D M , (i = 1, . . . , N),
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Also, by the hypothesis (H2 ), Bi vi = αi
N ∂vi (t, x) + v (t, x) ≥ K i j (x, y)v j (t, y)dy + εeγ ci N Ni t + h i (t, x) i ∂vi Ω j =1 N ≥ K i j (x, y)v j (t, y)dy + h i (t, x). Ω j =1
v
That is, u and satisfy the inequalities in (2.2), with v replaced by v . Therefore, by Step 1, u(t, x) < v (t, x) on D M . Taking ε → 0 yields the desired result and this completes the proof. Remark 2.1. Theorem 2.1 extends the corresponding results in [7–9] and [13]. The upper and lower solutions to system (1.1) are defined as follows. Definition 2.3. A vector function
u = (
u 1 , . . . ,
u N ) ∈ C 1,2(D M ) ∩ C(D M ) is called an upper solution of system (1.1) u satisfies on D M if
∂
ui u i ) in D M , ≥ fi (t, x,
u, ∇
u i , ∇ 2
∂t N Bi
K i j (x, y)
u j (t, y)dy on Γ M , ui ≥ Ω j =1
u i (t, x) ≥ u i0 (x)
in Ω , (i = 1, . . . , N).
A lower solution is defined analogously by reversing the above inequalities. As an immediate consequence of Theorem 2.1, we have: Corollary 2.1. Let the hypotheses (H1), (H2) and (H3) hold. Suppose that
u, u ∈ (C 1,2 (D M ) ∩ C 0,1 (D M )) N are upper and lower solutions of system (1.1) and u is any solution of system (1.1), then, u ≤ u ≤
u on D M . Corollary 2.1 concerns the order-preserving property of the upper and lower solutions. Moreover, it is easy to get the following corollary on the uniqueness of a solution to system (1.1). Corollary 2.2. Suppose system (1.1) has a solution and the hypotheses (H1), (H2 ) and (H3) hold. Then the solution for system (1.1) is unique. Example 2.1. Consider the following coupled system of convection–reaction–diffusion equations: ∂u i − L i (t)u i = gi (t, x, u, ∇u i ) (t > 0, x ∈ Ω ), ∂t N Bi u i = K i j (x, y)u j (t, y)dy (t > 0, x ∈ ∂Ω ),
(2.4)
Ω j =1
u i (t, x) = u i0 (x)
in Ω , (i = 1, . . . , N),
where the elliptic operator L i (t) is given by L i (t) =
n j,k=1
(i)
a j k (t, x)
n ∂2 ∂ (i) + b j (t, x) , ∂ x j ∂ xk ∂ xj j =1
(i = 1, . . . , N).
In view of Theorem 2.1, we obtain the following new information about the coupled system (2.4): Assume for each i, j = 1, . . . , N, the kernel K i j (x, y) satisfies the hypothesis (H2), function gi (·, u 1 , . . . , u N , ·) is nondecreasing with respect to u j ( j = i ) and satisfies the one-side Lipschitz condition gi (t, x, u, P) − gi (t, x, v, P) ≤ di
N (u i − vi ), j =1
if u ≥ v,
R.-N. Wang et al. / Applied Mathematics Letters 19 (2006) 1272–1277
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where u, v ∈ R N , P is any vector in Rn , and di > 0 is a constant. Suppose
u, u ∈ (C 1,2 (D M ) ∩ C 0,1 (D M )) N are upper and lower solutions of system (2.4) and u is any solution of system (2.4); then, u ≤ u ≤
u and u is a unique solution. Acknowledgments The authors thank Prof. C. V. Pao very much for sending us reprints of [8,9], which were helpful to us. Moreover, the authors are grateful to the anonymous referee for his/her careful reading and valuable suggestions. References [1] S. Carl, V. Lakshmikantham, Generalized quasilinearization method for reaction–diffusion equations under nonlinear and nonlocal flux conditions, J. Math. Anal. Appl. 271 (2002) 182–205. [2] W.A. Day, Extensions of a property of heat equation to linear thermoelasticity and other theories, Quart. Appl. Math. 40 (1982) 319–330. [3] W.A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math. 40 (1983) 468–475. [4] L.C. Evans, Partial differential equations, in: Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. [5] O.A. Ladyzenskaja, V. Solonnikov, N. Uralceva, Linear and quasilinear equations of parabolic type, Amer. Math. Soc. Transl. (1968). [6] D.G. Levitt, Interpretation of biological ion channel flux data: reaction rate versus continuum theory, Annu. Rev. Biophys. Chem. 15 (1986) 29–57. [7] C.V. Pao, Dynamics of reaction–diffusion equations with nonlocal boundary conditions, Quart. Appl. Math. 50 (1995) 173–186. [8] C.V. Pao, Dynamics of weakly coupled parabolic systems with nonlocal boundary conditions, in: Advances in Nonlinear Dynamics, in: Stability Control Theory Methods Appl., vol. 5, Gordon and Breach, Amsterdam, 1997, pp. 319–327. [9] C.V. Pao, Asymptotic behavior of solutions of reaction–diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math. 88 (1998) 225–238. [10] M.F. Schumaker, Boundary conditions and trajectories of diffusion process, J. Chem. Phys. 117 (2002) 2469–2473. [11] S. Seo, Global existence and decreasing property of boundary values of solutions to parabolic equations with nonlocal boundary conditions, Pacific. J. Math. 193 (2000) 219–226. [12] H.M. Yin, On a class of parabolic equations with nonlocal boundary conditions, J. Math. Anal. Appl. 294 (2004) 712–728. [13] Y.F. Yin, On nonlinear parabolic equations with nonlocal boundary condition, J. Math. Anal. Appl. 185 (1994) 54–60.