Solvability and stability of a semilinear wave equation with nonlinear boundary conditions

Nonlinear Analysis 73 (2010) 1495–1514

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Solvability and stability of a semilinear wave equation with nonlinear boundary conditions Andrzej Nowakowski ∗ Faculty of Mathematics, University of Lodz, Banacha 22, 90-238 Lodz, Poland University of Humanity and Economics, Sterlinga 26, 90-360 Lodz, Poland

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Article history: Received 3 March 2010 Accepted 22 April 2010 Keywords: Semilinear equation of the vibrating string Nonlinearity on the boundary Dual variational method Stability of solutions

We discuss solvability for the semilinear equation of the vibrating string xtt (t , y) − ∆x(t , y) + f (t , y, x(t , y)) = 0 in a bounded domain, and certain type of nonlinearity on the boundary. To this effect we derive a new dual variational method. Next we discuss stability of solutions with respect to initial conditions. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Throughout this paper, Ω will be a general open bounded domain in Rn , n ≥ 2, with a boundary Γ , assumed to be sufficiently smooth. The aim of this paper is to study an existence question over a finite interval [0, T ], T < ∞ (T -arbitrary but fixed). We consider the second order hyperbolic semilinear problem with nonlinear boundary condition containing dissipation and sources terms: , xtt (t , y) − ∆x(t , y) + l(t , y, x(t , y)) = 0,

in (0, T ) × Ω , ∂ν x(t , y) + x(t , y) + g (xt (t , y)) = h(x(t , y)) on Σ = (0, T ) × Γ , x(0, ·) = x (·) ∈ H (Ω ), 0

1

(1)

xt (0, ·) = x (·) ∈ L2 (Ω ). 1

The problems of that type were studied by many authors (see e.g. [1,2] and references therein) mainly with nonlinearities that do not depend on (t , y) and of special type. The main goal of those papers was twofold: (i) to study the well-posedness of the system given by (1) on the finite energy space, i.e. H 1 (Ω ) × L2 (Ω ), and (ii) to derive uniform decay rates of the energy when t → ∞. The well-posedness included existence and uniqueness of both local and global solutions. However, as it is shown in [1], if we want to admit more general nonlinearities then we cannot expect well-posedness in Hadamard sense. The main difficulty and the novelty of the problem considered is related to the presence of the boundary nonlinear term h(x). This difficulty has to do with the fact that Lopatinski condition does not hold for the Neumann problem. The above translates into the fact that in the absence of the damping, the linear map h → (x(t ), xt (t )), is not bounded from L2 (Σ ) → H 1 (Ω ) × L2 (Ω ), unless the dimension of Ω is equal to one. More details on this problem is discussed in [1,2]. The aim of this paper is to study a global existence question over a finite interval [0, T ], T < ∞ (T -arbitrary but fixed) for different type of nonlinearities l, g, h, than in [1,2] (in some cases less general) and so we are not interested in well-posedness in Hadamard sense. However, we still want to study stability of the system with respect to initial conditions i.e. continuous dependence (in some new sense) with respect to initial conditions.

∗

Corresponding address: Faculty of Math, University of Lodz, Banacha 22, 90-238 Lodz, Poland. Tel.: +48 42 6355961. E-mail address: [email protected].

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.04.035

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It is worth noting here that the nonlinearities l, h generally, may drive the solution of (1) to blow up in finite time [3–7] (see also discussion in [1] in context of appearance of the dissipation g). We exclude such a case imposing hypothesis Assumptions2, which relates some interaction between growth of l and length of time T . The importance of the problem with nonlinearity on the boundary appears in optimal control theory see e.g. [8–10,1] and the references therein. The problems like (1) where studied mostly by topological method, semigroup theory or monotone operator theory (see [1] for discussion about that). As it is known to the author there is no papers studying (1) (with nonlinearity on the boundary) by variational method. We would like to stress that just using the variational approach the dissipation term g on the boundary has no influence on the existence of solution to (1). This is why we do not discuss the interaction between dissipation and source on the boundary as it is usual done while approaching other methods (see e.g. [1,2]). What is essential in the method used here is that H = 12 x2 − H , where H is antiderivative of h, is convex function. Convexity is exploited in the paper in different forms and just convexity eliminates other assumptions. The general nonlinearity of interior source has the form: l(t , y, x) = j1 Fx1 (x) + j2 Fx2 (x) + · · · + jm Fxm (x) + G(t , y),

x ∈ R, (t , y) ∈ (0, T ) × Ω

where F , . . . , F are C , convex functions, G ∈ L (0, T ; L2 (Ω )) and j1 , . . . , jm is a sequence of numbers having values either −1 or +1. It turns out that in this case a size of initial conditions is not essential, however some restrictions on initial conditions are hidden in Assumptions2 below. We shall study (1) by variational method, i.e. we shall consider (1) as the Euler–Lagrange equations of two functionals: 1

m

J (x) =

T

Z

Z  Ω

0

Z

JΓ (x) =

1

0

T

Z Γ

1 2

1

|∇ x(t , y)|2 −

1 2


|xt (t , y)|2 + L(t , y, x(t , y)) dydt − x (T , ·) , x1 (·) L

2 (Ω )

,

(H (x(t , y))) + H ∗ (−∂ν x(t , y) − g (xt (t , y))) − hx(t , y), −∂ν x(t , y) − g (xt (t , y)i)dydt

(2)

(3)

defined on some subspaces of the space C ([0, T ]; H 1 (Ω )) and C ([0, T ]; H 1 (Γ )), respectively, discussed below. L is antiderivative of l with respect to the third variable and H ∗ is the Fenchel conjugate to H. Our purpose is to investigate (1) by studying critical points of functionals (2) and (3) or their some modification. To this effect we apply a new duality approach. As it is easily to see, functionals (2) and (3) are unbounded in C ([0, T ]; H 1 (Ω )) and this is a reason for which we are looking for critical points of J of ‘‘min–max’’ type. Our aim is to find a nonlinear subspace X of C ([0, T ]; H 1 (Ω )) and study (2) just only on X . In our approach the main difficulty is just the construction of the set X (compare [18]). In last section we prove stability of solutions of (1) when x0 (·) and x1 (·) are changing in suitable way. We will study the equation

∂ν x(t , y) + x(t , y) + g (xt (t , y)) = h(x(t , y)) on (0, T ) × Γ , x(0, ·) = x0 (·) ∈ H 1 (Γ ),

(4)

with the help of the functional JΓ (x) (see (3)) — Section 3 and we will investigate the equation xtt (t , y) − ∆x(t , y) + l(t , y, x(t , y)) = 0, in (0, T ) × Ω , x(t , y) = xΓ (t , y) on (0, T ) × Γ , xΓ (0, ·) = x0 (·) on Γ , 0 1 1 x(0, ·) = x (·) ∈ H (Ω ), xt (0, ·) = x (·) ∈ L2 (Ω ),

(5)

with the help of the functional J (x) (see (2)) — Section 4. 2. Main results We will focus on the case when n ≥ 3 (n = 2 being the least interesting, since the concept of criticality of Sobolev’s embedding is much less pronounced). First we formulate results concerning problem (4), its relation to the functional JΓ (x) and problem (5). To this effect we need to explain what do we mean by x0 (·) ∈ H 1 (Γ ) in (4), x0 (·) ∈ H 1 (Ω ) in (5) and the normal derivative ∂ν x(t , y) in (4) and in (3). Let us note that, by [11], each x0Γ ∈ H 1 (Γ ) can be ‘‘extended’’ inside Ω by a function x0 (·) ∈ H 1 (Ω ) such that Tx0 (·) = x0Γ ∈ H 1 (Γ ) where T is the trace operator on H 1 (Ω ). The same is true for each xΓ ∈ C ([0, T ]; H 1 (Γ )), i.e. each xΓ ∈ C ([0, T ]; H 1 (Γ )) can be ‘‘extended’’ inside Ω by a function x(·, ·) ∈ C ([0, T ]; H 1 (Ω )). ¯ in such a way that its weak Fix any t ∈ [0, T ], then x(t , ·), by [12], can by extended to Ex in some neighborhood of Ω derivative (in Sobolev sense) is well defined on Γ = ∂ Ω . Therefore by the normal derivative ∂ν x(t , y) in (4) and also in (3) we mean normal derivative of a function x(·, ·) ∈ C ([0, T ]; H 1 (Ω )) which is obtained from xΓ ∈ C ([0, T ]; H 1 (Γ )) in the way described above. In what follows we will always understand the functions from C ([0, T ]; H 1 (Γ )), C ([0, T ]; H 1 (Ω )) and normal derivative in the way described above and we do not use the different notations for functions from C ([0, T ]; H 1 (Γ )) and C ([0, T ]; H 1 (Ω )). Thus let x0 ∈ H 1 (Γ ). Instead of to investigate the functional JΓ (x) on the set UΓ =



x : x ∈ C ([0, T ]; H 1 (Γ )),

∂x ∈ C ([0, T ]; L2 (Γ )), x(0, y) = x0 (y), y ∈ Γ ∂t

we use a set Ut =



x : x(t , ·) = x0 (·) +

t

Z 0

 w(·)ds, w ∈ H 1 (Γ ) ,

t ∈ [0, T ]



A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

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and the functional JΓt (x) =

Z Γ

(H (x(t , y)) + H ∗ (−∂ν x(t , y) − g (t , xt (t , y))) − hx(t , y), −∂ν x(t , y) − g (t , xt (t , y))i)dy,

t ∈ [0, T ], (6)

considered on U t , t ∈ [0, T ]. That means we are looking for solutions to (4) in U t , t ∈ [0, T ]. Therefore, if such a solution belonging to U t , t ∈ [0, T ], exists it is a strong solution to (4). Assumptions concerning problem (4): (HΓ ) h(x) is Borel measurable for x ∈ R, x → H (x) = 12 x2 − H (x), where H is antiderivative of h, is convex and h satisfies growth condition a1 ≤ h(x) ≤ a2 |x|q−1 + b2 ,

x∈R

for some ai , b2 ∈ R, i = 1, 2, a2 > 0, and q > 2. g (·, s) is measurable in (0, T ), s ∈ R, g (t , ·) is continuous in R and satisfies growth condition g (t , s) ≥ a(t ) |s|r + b(t ),

(t , s) ∈ (0, T ) × R 2q−2 for some a, b ∈ L∞ (0, T ), a ≥ 0, b ≥ 0 and 0 ≤ r < q , moreover g (t , s) = −g (t , −s) for (t , s) ∈ (0, T ) × R. Proposition 1. x¯ (t , ·) ∈ U t , t ∈ [0, T ] is a solution to (4) if and only if x¯ (t , ·) affords a minimum to the functional JΓt defined on U t , t ∈ [0, T ] and JΓt (¯x) = 0. Proposition 2. Under assumption (HΓ ) there is an x¯ Γ (t , ·) ∈ U t which affords a minimum to the functional JΓt defined on U t , t ∈ [0, T ] and JΓt (¯x) = 0. Corollary 3. The problem (4) has a solution in U t , t ∈ [0, T ]. Remark 4. It is worth to note that the damping term g may be zero i.e. it has not influence on existence of solutions to (4) in U t , t ∈ [0, T ]. Usually (see [1,2,13]), damping term g is monotonic and source term h is smooth but not necessary monotonic, in our case we have, on the contrary, x − h monotonic (in case when term x is absent then simply h is monotonic) and dissipation term g is only continuous. The reason is that we use quite different approach to solve (1). We would like to stress, once more, that just convexity of x → H (x) = 12 x2 − H (x) is very important here and play a crucial role in the proof — it eliminates a role of damping term g. Just different method used here causes that it is difficult to compare assumptions concerning problem (1). In our case solutions of (4) are independent of solutions of (5). It is necessary to stress that solution to (4) exists on each finite interval [0, T ]. We introduce the definition of a weak solution to (5): Definition 5 (Weak Solution). By a weak solution to (5), defined on some interval [0, T ], we mean a function x ∈ U, where

∂x ∈ C ([0, T ]; L2 (Ω )), x(0, y) = x0 (y), ∂t  xt (0, y) = x1 (y), y ∈ Ω , x(t , y) = x¯ Γ (t , y), (t , y) ∈ Σ ,

 U =

x : x ∈ C ([0, T ]; H 1 (Ω )),

such that for all ϕ ∈ Uϕ T

Z 0

Z Ω

(−xt ϕt + ∇ x∇ϕ)dydt +

Z 0

T

Z Ω

lϕ dydt = −

Z Ω

xt (T , y)ϕ 0 (y)dy +

Z Ω

xt (0, y)ϕ(0, y)dy,

where

 ∂ϕ 0 Uϕ = ϕ : ϕ ∈ C ([0, T ]; H (Ω )), ∈ C ([0, T ]; L2 (Ω )), ϕ(T , y) = ϕ (y), y ∈ Ω , ϕ(t , y) = 0, (t , y) ∈ Σ . ∂t 

1

We shall consider U with a topology induced by the norm kxkU = (kxk2C ([0,T ];H 1 (Ω )) + kxt k2C ([0,T ];L2 (Ω )) )1/2 and by a ball

in U we mean a ball induced from C ([0, T ]; H 1 (Ω )).

Remark 6. The weak solution considered in this paper is stronger that e.g. in [2] where x ∈ Cw ([0, T ]; H 1 (Ω )), ∂∂ xt ∈ C ([0, T ]; L2 (Ω )) (Cw ([0, T ]; Y ) denotes the space of weakly continuous functions with values in a Banach space Y ). The main contribution of this paper is not a uniqueness results but global (for given finite interval [0, T ]) existence theorem to (1) and continuous dependence of solutions with respect to initial data in the following sense (because of lack of uniqueness): Definition 7. For given sequences x0n ⊂ H 1 (Ω ), x1n ⊂ L2 (Ω ), converging to x¯ 0 in H 1 (Ω ), x¯ 1 in L2 (Ω ), respectively, there





 

is a subsequence of {xn }-solutions to (1) corresponding to x0n , x1n , which we denote again by {xn } weakly convergent in H 1 ((0, T ) × Ω ) and strongly in L2 ((0, T ) × Ω ) to an element x¯ ∈ U being a solution to (1) corresponding to x¯ 0 , x¯ 1 .

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A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

To formulate assumptions concerning Eq. (5) we need to recall some theorem from [14]-linear case of (5) which we use as a starting point to study (5). Theorem 8. Let Q ∈ L1 (0, T ; L2 (Ω )), x¯ Γ ∈ H 1 (Σ ) is a solution to (4), x0 ∈ H 1 (Ω ), x1 ∈ L2 (Ω ) with the compatibility condition: x¯ Γ (0, y) = x0 (y), y ∈ Γ . Then there exists x being a unique weak solution to xtt (t , y) − ∆x(t , y) = Q (t , y), x(0, y) = x0 (y),

xt (0, y) = x1 (y),

x(t , y) = x¯ Γ (t , y),

(t , y) ∈ Σ

y ∈ Ω,

(7)

and such that x ∈ C ([0, T ]; H 1 (Ω )), xt ∈ C ([0, T ]; L2 (Ω )), ∂ν x¯ ∈ L2 (Σ ),



kxkC ([0,T ];H 1 (Ω )) ≤ C (kQ kL1 (0,T ;L2 (Ω )) + k¯xΓ kH 1 (Σ ) + x0 H 1 (Ω ) + x1 L (Ω ) ), 2

0

1



kxt kC ([0,T ];L2 (Ω )) ≤ D(kQ kL1 (0,T ;L2 (Ω )) + k¯xΓ kH 1 (Σ ) + x H 1 (Ω ) + x L (Ω ) ) 2

with some C > 0, D > 0 independent on Q . Shortly the solution x from the above theorem may be estimated by

kxkC ([0,T ];H 1 (Ω )) ≤ C kQ kL1 (0,T ;L2 (Ω )) + E w ,

t ∈ (0, T ),

(8)

kxt kC ([0,T ];L2 (Ω )) ≤ D kQ kL1 (0,T ;L2 (Ω )) + Aw , t ∈ (0, T ),   







where E w = C k¯xΓ kH 1 (Σ ) + x0 H 1 (Ω ) + x1 L (Ω ) and Aw = D k¯xΓ kH 1 (Σ ) + x0 H 1 (Ω ) + x1 L

2 (Ω )

2

(9)



and in the paper

we will just use the last estimations. Everywhere below constants C and D will always denote those occurring in (8), (9). Assumptions2 concerning Eq. (1). Let F 1 , F 2 , . . . , F m of the variable x and a function G of the variable (t , y) be given. F 1 , F 2 , . . . , F m are continuously differentiable and convex in R and satisfy F i (x) ≥ ai x + bi , for some ai , bi ∈ R, i = 1, . . . , m, for all x ∈ R, G (·, ·) ∈ L1 (0, T ; L2 (Ω )). Let j1 , . . . , jm be a sequence of numbers having values either −1 or +1. Assume that our original nonlinearity (see (5)) has the form l = j1 Fx1 + j2 Fx2 + · · · + jm Fxm + G.

(10)

There exists a ball F in U with center in zero, such that

Fxi

(x) ∈ L (0, T ; L2 (Ω )), i = 1, . . . , m, for x ∈ F and Fxm (x) L1 (0,T ;L 1

2 (Ω ))

is bounded in F. Moreover there exists some xˆ ∈ F and Em−1 such that



xˆ

C ([0,T ];H 1 (Ω ))


≤ C l xˆ L1 (0,T ;L

2 (Ω ))

+ Ew

(11)

and





j1 F 1 + j2 F 2 + · · · + jm−1 F m−1 xˆ 1 x x x L (0,T ;L

2 (Ω ))

≤ Em−1 .

(12)

Define the sets

n

m XFG = x ∈ F : kxkC ([0,T ];H 1 (Ω )) ≤ C (kl (x)kL1 (0,T ;L2 (Ω )) ) + E w ,

o




j1 F 1 + j2 F 2 + · · · + jm−1 F m−1 (x) 1 ≤ Em−1 , x x x L (0,T ;L2 (Ω ))  n
X mFG = v ∈ F : kvkC ([0,T ];H 1 (Ω )) ≤ C j1 Fx1 + j2 Fx2 + · · · + jm−1 Fxm−1 (v) L1 (0,T ;L (Ω )) 2  o

m w m

+ Fx (x) L1 (0,T ;L (Ω )) + kG (·, ·)kL1 (0,T ;L2 (Ω )) + E , x ∈ XFG . 2

(13)

(14)

m By Assumptions2 XFG and X mFG are nonempty.

Theorem 9 (Main Theorem). Under Assumptions2 there exists x ∈ X mFG such that J (x) = infx∈X m J (x) and x is a weak solution FG m to (5). Moreover x¯ ∈ XFG . Remark 10. We would like to stress that the result obtained in the paper is of global type — for each given, but fixed T < ∞. The known results concerning existence of global solution to (1) (without uniqueness and damping term) assume that nonlinearity l is locally Lipschitz and sublinear at infinity or of polynomial growth |s|p−1 s (p such that H 1 (Ω ) ⊂ Lp+1 (Ω ))

A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

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with additional boundedness for initial data (see e.g. [1]). We do not assume that l is locally Lipschitz from H 1 (Ω ) → Lp+1 (Ω ), as x → l(x) is only continuous in R. Instead, we assume the special structure of l (see (10)) — a finite combination i 1 (with coefficients

being −1 or +1) of convex functions. Moreover the assumptions: ‘‘Fx (x) ∈ L (0, T ; L2 (Ω )), i = 1, . . . , m for x ∈ F and Fxm (x) L1 (0,T ;L (Ω )) is bounded in F’’ imply growth conditions for l according to the Sobolev’s embedding 2

H 1 (Ω ) → L

2n n−2

(Ω ). Some type of boundedness for initial data are hidden in inequalities (11), (12) and the definitions of the

m sets XFG , X mFG . We would like to stress that l is only continuous in x, usually l is at least of C 1 (see e.g. [1,2,13]). Moreover we are looking for solutions in bounded set X mFG , so all solutions are bounded in the norm of C ([0, T ]; H 1 (Ω )). This is a case which is often assumed to get global solution (see [1,2] and references therein).

We have no uniqueness of solutions to (1), but a kind of continuous dependence on initial data is still possible. Theorem 11. Assume (HΓ ) and Assumptions2. Let x0n , x1n given sequences in H 1 (Ω ), L2 (Ω ) respectively, converging to x¯ 0 ,

 

x¯ 1 in H 1 (Ω ), L2 (Ω ). Then there is a subsequence of {xn }-solutions to (1) corresponding to x0n , x1n , which we denote again by {xn } weakly convergent in H 1 ((0, T )×Ω ) and strongly in L2 ((0, T )×Ω ) to an element x¯ ∈ U being a solution to (1) corresponding to x¯ 0 , x¯ 1 .

 

Let F i∗ be the Fenchel conjugate of F i i = 1, . . . , m − 1, F¯ m∗ the Fenchel conjugate of F¯ m = jm F m + xG. Define a dual to J functional JDmFG : L2 (0, T , L2 (Ω )) × L2 (0, T , L2 (Ω )) × · · · × L2 (0, T , L2 (Ω )) → R, as: JDmFG (p, q, z1 , . . . , zm−1 ) = −j1

T

Z 0

T

Z − j2

Ω

0

+

1 2

T

Z 0

Z Z Ω

Z Ω

F 1∗ (t , y, (pt (T − t , y) + div q (t , y) + z1 (t , y) + · · · + zm−1 (t , y))) dydt

F 2∗ (t , y, z1 (t , y))dydt − · · · −

T

Z 0

Z Ω

F¯ m∗ (t , y, zm−1 (t , y))dydt −


|p (T − t , y)|2 dydt + x0 (·) , p (T , ·) L

2

Z − (Ω )

Σ

1 2

Z 0

T

Z Ω

|q (t , y)|2 dydt

x¯ Γ (t , y) (q (t , y) , ν(y)) dydt .

(15)

Now we can formulate theorem which gives us additional information on solutions to (1) important in classical mechanics. This theorem is absolutely new for problem (1). Theorem 12 (Variational Principle and Duality Result). Assume Assumptions2. Let x ∈ X mFG be such that J (x) = infx∈X m J (x). FG Then there exists (¯p, q¯ ) ∈ L2 (0, T , L2 (Ω )) × L2 (0, T , L2 (Ω )) such that for a.e. (t , y) ∈ (0, T ) × Ω , p¯ (T − t , y) = x¯ t (t , y),

(16)

q¯ (t , y) = ∇ x¯ (t , y),

(17)

p¯ t (T − t , y) + div q (t , y) − l (t , y, x¯ (t , y)) = 0

(18)

and J (¯x) = JDmFG (¯p, q¯ , z¯1 , . . . , z¯m−1 ) , where z¯i = −ji+1 Fxi+1 (t , y, x¯ (t , y)) ,

i = 1, . . . , m − 2,

z¯m−1 = −F¯xm (t , y, x¯ (t , y)) .

(19)

m Moreover x¯ ∈ XFG .

The proofs of theorems are given in Sections 3 and 4. They consist of several steps. First we prove Propositions 1 and 2 and Corollary 3, i.e. we solve problem (4). Next we prove Theorem 9 (Main theorem). First for nonlinearity l consisting only of one function j1 Fx1 + G, next for difference of two Fx1 − Fx2 + G and then by induction for general case. 3. Proof of existence for problem (4) As mentioned before, the main difficulty of the problem under study is the fact that the Neumann problem does not satisfy Lopatinski condition and therefore, the map from the boundary data in L2 (Σ ) into finite energy space is not bounded (unless dimension of Ω is equal to one). In order to cope with this problem in [1,2] a regularizing term — strongly monotone dissipation is introduced, whose effect is to ‘force’ the Lopatinski condition. We follow, in quite, different way. We convert the problem (4) into variational one and this allow us to omit the meaning of the dissipation term. However, the price we pay for that is the type of nonlinearity for the boundary source term h (see (HΓ )). Being inspired by Brezis–Ekeland [15] (see also [16]) we formulate the variational principle for problem (4). Proposition 13. x¯ ∈ U Γ is a solution to (4) if and only if x¯ affords a minimum to the functional JΓ defined on U Γ and JΓ (¯x) = 0.

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A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

Proof. It is a simple consequence of the equivalence of the following two relations in (0, T ) × Γ :

(i) − ∂ν x¯ (t , y) − g (t , x¯ t (t , y)) ∈ ∂ H (t , y, x¯ (t , y)), (ii) H (¯x(t , y)) + H ∗ (−∂ν x¯ (t , y) − g (t , x¯ t (t , y))) = h¯x(t , y), −∂ν x¯ (t , y) − g (t , x¯ t (t , y))i and the inequality H (x(t , y)) + H ∗ (−∂ν x(t , y) − g (t , xt (t , y))) ≥ hx(t , y), −∂ν x(t , y) − g (t , xt (t , y))i which holds for all x ∈ U Γ .



Let us notice that the sets U t are nonempty. Instead of consider the functional (3) we will study the functional JΓt (x) on U t , t ∈ [0, T ] and suitable modification of the above proposition i.e. Proposition 1, which proof we omit. We prove under hypothesis (HΓ ) that there exists a minimum to the functional JΓt defined on U t , t ∈ [0, T ], i.e. the proof of Proposition 13. Proof of Proposition 13. Let us notice that JΓt (x) is bounded below in U t , t ∈ [0, T ], in fact, JΓt (x) ≥ 0 and weakly lower semicontinuous in U t , t ∈ [0, T ]. Moreover, let us observe, by (HΓ ), JΓt (x) → ∞ when kxk 1, q → ∞. Really, it is W

enough to notice that for t ∈ [0, T ]

Z Γ

Z Γ

H (x(t , y))dy ≥

Z Γ

q−1

(Γ )

(a1 |x(t , y)|2 + b1 )dy,

H ∗ (−∂ν x(t , y) − g (t , xt (t , y)))dy ≥

(20)

q ∂ν x(t , y) + g (t , xt (t , y)) q−1 − b2 dy a !

Z Γ

a2 (1 − 1/q)

2

Z   q −1 ≥ (a2 ) q−1 (1 − 1/q) a(t ) |xt (t , y)|r + b(t ) − |∂ν x(t , y)| q−1 − b2 dy ZΓ  
q −1 ≥ (a2 ) q−1 (1 − 1/q) |∂ν x(t , y)| − a(t ) |xt (t , y)|r + b(t ) q−1 − b2 dy Γ

  q

q −1 ≥ (a2 ) q−1 (1 − 1/q) k∇ x(t , ·)νkLq−q1 − a(t ) |xt (t , ·)|r + b(t ) Lq−q1 . q−1

(21)

q−1

From (20) and (HΓ ) we infer that kxn kL2 (Γ ) is bounded for minimizing sequence {xn } of JΓt and next from (21) that

k∇ xn kL 1,

q q−1

(Γ )

is bounded. Thus, there is a subsequence of {xn } (which we again denote by {xn }) such that it is weakly in

q

W q−1 (Γ ) convergent to some x¯ Γ and pointwise convergent to it. Therefore, for each t ∈ [0, T ], lim infn→∞ JΓt (xn ) ≥ JΓt (¯x). Now let us define the dual functional to JΓt , t ∈ [0, T ], by JΓt D (x) =

Z Γ

(H (−x(t , y)) + H ∗ (∂ν x(t , y) + g (t , xt (t , y))) − hx(t , y), ∂ν x(t , y) + g (t , xt (t , y))i)dy,

t ∈ [0, T ]. (22)

It is clear that JΓt (x) = JΓt D (−x) for all x ∈ U t and so infx∈U t JΓt (x) = infx∈U t JΓt D (x), t ∈ [0, T ]. By the duality theory for convex functionals (see [17,16]) we have that infx∈U t JΓt (x) = − infx∈U t JΓt D (x), t ∈ [0, T ]. Hence we infer that JΓt (¯xΓ ) = 0.  4. Proof of existence for problem (5) 4.1. Simply case — one function: l = Fx First consider another equation (with x¯ Γ ∈ H 1 (Σ ) being a solution to (4)) xtt (t , y) − ∆x(t , y) + Fx (t , y, x(t , y)) = 0, in (0, T ) × Ω , x(t , y) = x¯ Γ (t , y), (t , y) ∈ Σ , x(0, ·) = x0 (·) ∈ H 1 (Ω ), xt (0, ·) = x1 (·) ∈ L2 (Ω )

(23)

and corresponding to (23) functional J (x) = F

Z 0

T

Z  Ω

1 2

|∇ x(t , y)| − 2

1 2

|xt (t , y)|

2



T

Z

Z

dydt + 0

Ω

F (t , y, x (t , y)) dydt − x (T , ·) , x1 (·)




L2 (Ω )

(24)

defined on U. (23) is the Euler–Lagrange equation for the action functional J F . We shall use often the following notation: Hx (h) = Hx (·, ·, h(·, ·)). For that problem we assume the following hypotheses: G1. F (t , y, x) is measurable with respect to (t , y) in (0, T ) × Ω for all x in R and continuously differentiable and convex with respect to the third variable in R for a.e. (t , y) ∈ (0, T ) × Ω . Fx (t , y, x) = Fx1 (x) + F 2 (t , y), (t , y, x) ∈ (0, T ) × Ω × R, F 2 (·, ·) ∈ L1 (0, T ; L2 (Ω )).

A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

1501

G2. There exists constant E, such that for some xˆ ∈ U,


≤ C Fx1 xˆ L1 (0,T ;L



xˆ

C ([0,T ];H 1 (Ω ))

2



+ F 2 (·, ·) L1 (0,T ;L (Ω ))

+ Ew

2 (Ω ))

and Fx1 xˆ L1 (0,T ;L (Ω )) ≤ E . 2 G3




F (t , y, x) ≥ a(t , y)x + b(t , y), for some a, b ∈ L2 ((0, T ) × Ω ), for all x ∈ R. Define the sets: XF =

n



x ∈ U : kxkC ([0,T ];H 1 (Ω )) ≤ C Fx1 (x) L1 (0,T ;L (Ω )) 2

 o



+ F 2 (·, ·) L1 (0,T ;L (Ω )) + E w , Fx1 (x) L1 (0,T ;L (Ω )) ≤ E , 2 2 n 

X F = v ∈ U : kvkC ([0,T ];H 1 (Ω )) ≤ C Fx1 (x) L1 (0,T ;L (Ω )) + F 2 (·, ·) L1 (0,T ;L 2

(25)

 2 (Ω ))

o

+ E w for some x ∈ XF .

(26)

It is clear that XF ⊂ X F . The dual functional, now reads

(p, q) = −

JDF

T

Z

Z Ω

0

+

1

1

0


kp (T − t , ·)k2L2 (Ω ) dt + x0 (·) , p (T , ·) L

2

T

Z

2

T

Z

2

F (t , y, (pt (T − t , y) + div q (t , y))) dydt − ∗

0

Z − (Ω )

Σ

kq (t , ·)k2L2 (Ω ) dt x¯ Γ (t , y) (q (t , y) , ν(y)) dydt

(27)

where ν = (ν1 , . . . , νn ) is the unit outward normal to Γ , for a.e. t ∈ [0, T ], pt (T − t , ·) + div q (t , ·) is an element of L2 (Ω ) and F ∗ (t , y, ·) is the Fenchel conjugate to F (t , y, ·), i.e. F ∗ (t , y, h) = supd∈R {hd − F (t , y, d)}, for (t , y) ∈ [0, T ] × Ω , h ∈ R, hence T

Z

Z

0

Ω

F ∗ (t , y, (pt (T − t , y) + div q (t , y)))dydt T

Z =

Z

sup

v∈L2 ((0,T )×Ω )

0

Ω

(pt (T − t , y) + div q (t , y))v(t , y)dydt −

T

Z 0

Z Ω

F (t , y, v (t , y)) dydt



and JDw : UD =

  (p, q) : p ∈ C ([0, T ]; L2 (Ω )), pt (T − ·, ·) + div q (·, ·)
n ∈ → R. 1 1 L (0, T ; L2 (Ω )), p(0, ·) = x (·), q ∈ L2 (0, T ; L2 (Ω ) )

We prove the following Lemma 14. There exist constants C1w , C2w , C3w , C4w independent on x ∈ XF such that

kvkL2 (0,T ;H 1 (Ω )) ≤ C2w ,

kvt kL2 (0,T ;L2 (Ω )) ≤ C1w ,

kvtt kL2 (0,T ;H −1 (Ω )) ≤ C3w , k∆vkL2 (0,T ;H −1 (Ω )) ≤ C4w ,  

kvkC ([0,T ];H 1 (Ω )) ≤ C E + F 2 (·, ·) L1 (0,T ;L (Ω )) + E w , 2  

2

kvt kC ([0,T ];L2 (Ω )) ≤ D E + F (·, ·) L1 (0,T ;L (Ω )) + Aw 2

where v is the weak solution of the problem

vtt (t , y) − ∆v(t , y) = −Fx (t , y, x(t , y)) in (0, T ) × Ω , v(0, y) = x0 (y), vt (0, y) = x1 (y), y ∈ Ω , v(t , y) = x¯ Γ (t , y), (t , y) ∈ Σ ,

(28)

with x ∈ XF . Proof. Fix arbitrary x ∈ XF . Since x ∈ U and by the assumptions on F , see G2, it follows that Fx (·, ·, x(·, ·)) ∈ L1 (0, T ; L2 (Ω )). Hence by Theorem 8 and (8) there exists a unique solution v ∈ U of our problem for the Eq. (28) satisfying kvkC ([0,T ];H 1 (Ω )) ≤ C kFx (x)kL1 (0,T ;L2 (Ω )) + E w . Taking into account the definition of the set X F we get the following estimation, independent on

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A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

. Moreover, by (8) we get 

≤ TC kFx (x)kL1 (0,T ;L2 (Ω )) + E w ≤ TC E + F 2 (·, ·) L1 (0,T ;L

x ∈ XF , kFx (x)kL1 (0,T ;L2 (Ω ))

kvkL2 (0,T ;H 1 (Ω ))

≤ E + F 2 (·, ·) 1

L (0,T ;L2 (Ω ))



+ Ew,



+ Aw

2 (Ω ))

by (9)



kvt kL2 (0,T ;L2 (Ω )) ≤ TD kFx (x)kL1 (0,T ;L2 (Ω )) + Aw ≤ TD E + F 2 (·, ·) L1 (0,T ;L

2 (Ω ))

and since for some E1 : k∆vkL2 (0,T ;H −1 (Ω )) ≤ E1 kvkL2 (0,T ;H 1 (Ω )) thus





kvtt kL2 (0,T ;H −1 (Ω )) ≤ E1 TC E + F 2 (·, ·) L1 (0,T ;L

2 (Ω ))



+ E1 E w + E + F 2 (·, ·) L1 (0,T ;L

2 (Ω ))



/T .

Hence, putting







C1w = TD E + F 2 (·, ·) L1 (0,T ;L (Ω )) + Aw , 2



C2w = TC E + F 2 (·, ·) L1 (0,T ;L (Ω )) + E w , 2







C3w = E1 TC E + F 2 (·, ·) L1 (0,T ;L (Ω )) + E1 E w + C w /T 2

and



C4w = E1 TC E + F 2 (·, ·) L1 (0,T ;L (Ω )) + E1 E w 2

we infer the first assertion of the lemma. Further by (8) we get

kvkC ([0,T ];H 1 (Ω )) ≤ C kFx (·, ·, x(·, ·))kL1 (0,T ;L2 (Ω )) + E w . Now by (25) it follows that

 

kvkC ([0,T ];H 1 (Ω )) ≤ C E + F 2 (·, ·) L1 (0,T ;L (Ω )) + E w , 2  

2

kvt kC ([0,T ];L2 (Ω )) ≤ D E + F (·, ·) L1 (0,T ;L (Ω )) + Aw 2

and we get the last assertion of the lemma.



Proposition 15. For every x ∈ XF the weak solution e x of the problem

e xtt (t , y) − ∆e x(t , y) = −Fx (t , y, x (t , y)), e e x(0, y) = x0 (y), xt (0, y) = x1 (y), y ∈ Ω , e x(t , y) = x¯ Γ (t , y), (t , y) ∈ Σ

(29)

belongs to X F . Proof. Fix arbitrary x ∈ XF . Since x ∈ U and by the definition of the set X F , it follows that Fx (·, ·, x(·, ·)) ∈ L1 (0, T ; L2 (Ω )). Hence by Theorem 8 there exists a unique weak solution e x ∈ U of problem (29). Moreover e xtt − ∆e x ∈ L1 (0, T ; L2 (Ω )). By F the definition of the set X , it follows that



kFx (·, ·, x(·, ·))kL1 (0,T ;L2 (Ω )) ≤ E + F 2 (·, ·) L1 (0,T ;L

2 (Ω ))



.

Further by (8) we get

ke xkC ([0,T ];H 1 (Ω )) ≤ C kFx (·, ·, x(·, ·))kL1 (0,T ;L2 (Ω )) + E w and further that



ke xkC ([0,T ];H 1 (Ω )) ≤ C Fx1 (x) L1 (0,T ;L

2 (Ω ))

+ F 2 (·, ·) L1 (0,T ;L

Thus for an arbitrary x ∈ XF there exists an e x ∈ XF .

2 (Ω ))



+ Ew.



Remark 16. Proposition 15 apparently may suggest that it is more convenient to apply a suitable fixed point theorem in order to get the existence of weak solutions to problem (1). Indeed, the above mentioned proposition states that the map H assigning to x ∈ XF a weak solution e x ∈ X F of (29) has the property H (XF ) ⊂ X F . However this is only starting point in fixed point theory. In order to proceed with the so called topological method we must prove that the map H and set H (XF ) possess suitable properties. However the assumptions G1–G3 (especially (25)) do not imply directly (if at all) neither

A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

1503

that H is contraction nor that H (X F ) is convex and relatively compact. We have chosen variational approach which ensures not only the existence of solutions but also certain variational properties of solutions which are absolutely new in that case. Lemma 17. Each x ∈ X F has the derivatives xtt and ∆x belonging to L∞ (0, T ; H −1 (Ω )) and hence to L2 (0,nT ; o H −1 (Ω )). w Moreover, sets Xttw = xtt : x ∈ X F , X∆ = ∆x : x ∈ X F are bounded in L2 (0, T ; H −1 (Ω )). Hence each sequence xtt from Xttw







j



n o

has a subsequence converging weakly in L2 (0, T ; H −1 (Ω )) to a certain element from a set Xttw and the sequence xt is convergent j

strongly in L2 (0, T ; H −1 (Ω )). Proof. It is clear, by Lemma 14, that

n o xtt

from Xttw has a subsequence converging weakly in L2 (0, T ; H −1 (Ω )). By the

 j

is also weakly (up to some subsequence) converging in L2 (0, T ; H 1 (Ω )) to some

j

same lemma corresponding sequence x

n o j

element x¯ . By the definition of X F and uniqueness of weak solutions of (29) we infer that xtt

has subsequence weakly

 convergent to x¯ tt . The last, the fact that ∆xj is bounded in L2 (0, T ; H −1 (Ω )) and known theorem imply that for this n o j subsequent xt is convergent strongly in L2 (0, T ; H −1 (Ω )) to x¯ t .  We observe that functional J F is well defined on X F . Moreover by (G3) we have Lemma 18. There exist constants M1F , M2F such that M1F ≤

T

Z

Z Ω

0

F (t , y, x(t , y))dydt ≤ M2F

for all x ∈ X F . Lemma 19. The functional J F attains its minimum on XF in X F i.e. infx∈XF J F (x) = J F (x), where x ∈ X F .



Proof. By definition of the set XF and Lemma 18 we see that the functional J F is bounded in XF . We denote by xj a

 j

minimizing sequence for J F in XF . This sequence, by Lemma 14, has subsequence which we denote again by x converging weakly in L2 (0, T ;H 1 (Ω )) and strongly in L2 (0, T ; L2 (Ω )), hence also strongly in L2 ((0, T ) × Ω ; R) to a certain element x ∈ U. Moreover xj is also convergent almost everywhere. Thus by construction of the set X F we observe that x ∈ X F . Hence lim inf J F xj ≥ J F (¯x).




j→∞

Thus inf J F xj = J F (¯x).




x∈XF



To consider properly the dual action functional let us put Wt1F = p ∈ C ([0, T ]; L2 (Ω )) : pt ∈ L2 (0, T ; H −1 (Ω )), p(0, ·) = x1 (·)





and


n

Wy1F = q ∈ L2 (0, T ; L2 (Ω ) ) : there exists p ∈ Wt1F such that pt (T − ·, ·) + div q(·, ·) ∈ L2 (0, T ; L2 (Ω )) .





and define a set on which a dual functional will be considered. Definition of X Fd : We say that an element (p, q) ∈ Wt1F × Wy1F belongs to X Fd provided that there exists x ∈ XF such that for a.e. t ∈ (0, T )

− pt (T − t , ·) − div q (t , ·) = −Fx (t , ·, x(t , ·))

(30)

with q (t , ·) = ∇ x (t , ·) or else p (T − t , ·) = xt (t , ·)

with q (t , ·) = ∇ x (t , ·) .

(31)

Remark 20. The definition of X Fd says that for each x ∈ XF there exist in X Fd two pairs of (p, q): one defined by (30), second defined by (31). We observe that neither X F nor X Fd is a linear space. Thus even standard calculations using convexity arguments are rather difficult. What helps us is a special structure of the sets XF and X Fd which despite their nonlinearity makes these calculations possible.

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A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

Now we may state the first step to main result of the paper which is the following existence theorem. Theorem 21. There is x ∈ XF , that infx∈XF J F (x) = J F (x). Moreover, there is (p, q) ∈ X Fd such that JDF (p, q) = inf J F (x) = J F (x)

(32)

x∈XF

and the following system holds, for t ∈ [0, T ], xt (t , ·) = p (T − t , ·) ,

(33)

∇ x (t , ·) = q (t , ·) ,

(34)

−pt (T − t , ·) − div q (t , ·) = −Fx (t , ·, x (t , ·)) .

(35)

We illustrate, by an example, a case of the above theorem. Example 22. Let us consider F 1 (t , y, x) = x6 + x5 + x2 + 1. Of course F 1 (t , y, ·) is convex. Let F 2 (·, ·) be any function in (0, T ) × Ω bounded in the norm k·kL1 (0,T ;L2 (Ω )) by 1. Choose such T ≤ 1 that C = 1, Ω = B(0, 1) (ball at zero

with radius 1) and take such initial values and boundary conditions that E w = 2. Note that for xˆ (·, ·) = 1, xˆ U ≤




C ( Fx1 xˆ L1 (0,T ;L

+ F 2 (·, ·) 1

2 (Ω ))

L (0,T ;L2


) + E w . Take E = 10 then Fx1 xˆ L1 (0,T ;L (Ω ))

2 (Ω ))

< 10. Thus assumptions G1,

G2 and G3 are satisfied, so by the above theorem there exists x ∈ XF being a solution to (23). 4.1.1. Variational principle We state the necessary conditions. We observe that due to the construction of the set X F and by the second Remark following Proposition 15 it follows that a minimizing sequence in XF for the functional J F may be assumed to be weakly convergent in L2 (0, T ; H 1 (Ω )) and strongly in L2 (0, T ; L2 (Ω )). Theorem 23. Let infx∈XF J F (x) = J F (x), where x ∈ L2 (0, T ; L2 (Ω )) is a limit, strong in L2 (0, T ; L2 (Ω )) and weak in L2 (0, T ; H 1 (Ω )), of a minimizing sequence xj ⊂ XF . Then there exist (¯p, q¯ ) ∈ X Fd such that for a.e. t ∈ (0, T ),



p¯ (T − t , ·) = x¯ t (t , ·),

(36)

q¯ (t , ·) = ∇ x¯ (t , ·),

(37)

−¯pt (T − t , ·) − div q (t , ·) + Fx (t , ·, x¯ (t , ·)) = 0

(38)

and J F (¯x) = JDF (¯p, q¯ ) . Moreover x¯ ∈ XF . Proof. Let x¯ ∈ X F be such that J F (¯x) = infxj ∈XF J F xj . Define




− p¯ t (T − t , ·) = div q¯ (t , ·) − Fx (t , ·, x¯ (t , ·)) ,

t ∈ (0, T )

(39)

with q¯ given by q¯ (t , y) = ∇ x¯ (t , y) for all (t , y) ∈ (0, T ) × Ω . F

By the definitions of J , J (¯x) = F

Z

JDF ,

 T Z

1

Ω

2

0

T Z

Z ≤−

Ω

0

+

1 2

Z

0

Ω

1 2


¯ |¯xt (t , y)| + F (t , y, x (t , y)) dydt − x¯ (T , ·) , x1 (·) L 2

2 (Ω )

h¯xt (t , y) , p¯ (T − t , y)i dydt +

T Z

Z 0

Ω

=−

relations (40), (39) and the Fenchel–Young inequality it follows that

|∇ x¯ (t , y)|2 −

Z

0 T

Z

T

Z

(40)

|¯p (T − t , y)|2 dydt −

1 2

T

Z 0

Z Ω

Ω

h∇ x¯ (t , y) , q¯ (t , y)i dydt

|¯q (t , y)|2 dydt +

F ∗ (t , y, (¯pt (T − t , y) + div q¯ (t , y))) dydt −

+ x0 (·) , p¯ (T , ·)


L2

+ (Ω )

1 2

T

Z 0

Z Ω

T

Z 0

1

T

Z

2

|¯p (T − t , y)|2 dydt −

Z Ω

0

Z Σ

Z Ω

F (t , y, x¯ (t , y))dydt

|¯q (t , y)|2 dydt

x¯ Γ (t , y) (¯q (t , y) , ν(y)) dydt .

A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

1505

Therefore we get that J F (¯x) ≤ JDF (¯p, q¯ ) . Let p , q



j

j



⊂ X

(41)

 j

Fd

Fd

denote the sequences corresponding to x

accordingly to the definition of the set X . It is clear that

the above (p, q) is a limit of the sequence pj , qj ∈ X Fd . We observe that there are two possible forms for the sequence  j j  p , q ⊂ X Fd corresponding to the sequence xj accordingly to the definition of the set X Fd with qj = ∇ xj . Namely, for (t , y) ∈ (0, T ) × Ω , j = 1, 2, . . .





j

pj (T − t , y) = xt (t , y)

qj (t , y) = ∇ xj (t , y) ,

(42)

or

− pjt (T − t , ·) = div qj (t , ·) − Fx (t , ·, xj (t , ·)),

qj (t , y) = ∇ xj (t , y) .

(43)

 j

First we investigate the convergence of both sequences. As for the sequence (42) we obviously get, since x

n o j

strongly in C ([0, T ]; H 1 (Ω )), xt j

xt * xt = p ,

converges

n o j

weakly in L2 (0, T ; L2 (Ω )) and pt weakly in L2 (0, T ; H −1 (Ω ))

∇ xj → ∇ x = q.

Since xj converges pointwise to x therefore



Fx t , y, xj (t , y)

converges pointwise to Fx (t , y, x¯ (t , y))




too. Therefore by the Fenchel–Young equality we have J F (¯x) = inf J F (x) = lim inf J F (xj ) j→∞

x∈X F

T

Z

Z 

≥ lim inf j→∞

2

Ω

0

T

Z

1

Z

+ lim inf j→∞

Ω

0




1 j 2 xt (t , y) − xj (T , y) , x1 (y) dydt 2

|∇ xj (t , y)|2 −



F t , y, xj (t , y) dydt




Z TZ Z TZ j 1 p (T − t , y) 2 dydt h∇ x¯ (t , y) , q¯ (t , y)i dydt + lim inf |¯q (t , y)|2 dydt + j →∞ 2 0 Ω 2 0 Ω 0 Ω Z T D  Z TZ E
j + lim inf xj (t , ·) , pt (T − t , ·) dt + x0 (·) , pj (T , ·) L (Ω ) + F (t , y, x¯ (t , y)) dydt 1 −1

≥−

T

Z

1

Z

j→∞

≥−

T

Z

1 2

0 T

Z

Ω

Z

− Z0 − Σ

≥

JDF

H ,H

0

Z

Ω

1

|¯q (t , y)|2 dydt +

T

Z

2

Z Ω

0

2

|¯p (T −, y)|2 dydt +

T

Z 0

Z Ω

Ω

0

F (t , y, x¯ (t , y)) dydt

h¯x (t , y) , div q¯ (t , y) − p¯ t (T − t , y)i dydt

x¯ Γ (t , y) h¯q (t , y) , ν(y)i dydt + x0 (·) , p¯ (T , ·)




L2 (Ω )

(¯p, q¯ ) .

Inequality J F (¯x) ≥ JDF (¯p, q¯ ) and (41) imply equality J F (¯x) = JDF (¯p, q¯ ). Moreover J F (¯x) = JDF (¯p, q¯ ) , implies T

Z

Z Ω

0

2

T Z

Z

2

1

T

Z

Z

0

1

+ =

F (t , y, (¯pt (T − t , y) + div q¯ (t , y))) dydt + ∗

Ω

0 T

Z

Z Ω

Z0

− Σ

|q (t , y)|2 dydt +

|¯xt (t , y)|2 dydt +

1 2

1

T Z

Z 0 T

Z

2

Ω

Z Ω

0

Ω

F (t , y, x¯ (t , y)) dydt

|∇ x¯ (t , y)|2 dydt

|¯p (T − t , y)|2 dydt

x¯ Γ (t , y) (¯q (t , y) , ν(y)) dydt + x0 (·) , p¯ (T , ·)




L2 (Ω )

+ x¯ (T , ·) , x1 (·)




L2 (Ω )

.

Therefore by (40), (39) and standard convexity arguments 1 2

T

Z 0

Z Ω

|¯p (T − t , y)|2 dydt +

1 2

T

Z 0

Z Ω

|¯xt (t , y)|2 dydt =

T

Z 0

Z Ω

h¯xt (t , y) , p¯ (T − t , y)i dydt .

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A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

Hence by the Fenchel–Young transform we obtain that p¯ (T − t , y) = x¯ t (t , y) i.e. (36). From (40), (39) and (36) we also infer that x¯ ∈ XF .  4.2. Simply case — one function: l = −Fx A similar theorem is true for the problem xtt (t , y) − ∆x(t , y) − Fx (t , y, x(t , y)) = 0, in (0, T ) × Ω , x(t , y) = x¯ Γ (t , y), (t , y) ∈ Σ , x(0, ·) = x0 (·) ∈ H 1 (Ω ), xt (0, ·) = x1 (·) ∈ L2 (Ω )

(44)

and corresponding to it functional J

F−

( x) =

T

Z

Z 

1 2

Ω

0

|∇ x(t , y)| − 2

1 2




|xt (t , y)| − F (t , y, x(t , y)) dydt − x (T , ·) , x1 (·) L 2

2 (Ω )

,

(45)

defined on U with same hypotheses G1, G2, G3 and the sets XF , X F . Really, Lemmas 14 and 17–19 are still valid as sign of F does not change their proofs, also the proof of the above theorem does not change. Hence we get for (44) the following theorem. Theorem 24. There is x ∈ X F , that infx∈XF J F − (x) = J F − (x). There is (¯p, q¯ ) ∈ L2 (0, T ; L2 (Ω )) × L2 (0, T ; L2 (Ω )) such that for a.e. (t , y) ∈ (0, T ) × Ω , p¯ (T − t , y) = x¯ t (t , y),

(46)

q¯ (t , y) = ∇ x¯ (t , y),

(47)

−¯pt (T − t , y) − div q (t , y) − Fx (t , y, x¯ (t , y)) = 0

(48)

and J F − (¯x) = JDF − (¯p, q¯ ) , where T

Z

JDF − (¯p, q¯ ) =

Z Ω

0

+

1

T

Z

2

F ∗ (t , y, (¯pt (T − t , y) + div q¯ (t , y))) dydt −

Z Ω

0

|¯p (T − t , y)|2 dydt −

Z Σ

1 2

T

Z

Z

0

Ω

|¯q (t , y)|2 dydt

x¯ Γ (t , y) (¯q (t , y) , ν(y)) dydt + x0 (·) , p¯ (T , ·)




L2 (Ω )

.

Moreover x¯ ∈ XF . 4.3. The case of nonlinearity F − G Now we consider more complicated problem i.e. xtt (t , y) − ∆x(t , y) − Gx (t , y.x(t , y)) + Fx (t , y, x(t , y)) = 0, x(t , y) = x¯ Γ (t , y), (t , y) ∈ Σ , x(0, ·) = x0 (·) ∈ H 1 (Ω ), xt (0, ·) = x1 (·) ∈ L2 (Ω ),

in (0, T ) × Ω , (49)

and corresponding to it functional J

FG

(x) =

T

Z

Z 

2

Ω

0

T

Z

1

Z

+ 0

Ω

|∇ x(t , y)| − 2

1 2



|xt (t , y)| − G(x(t , y)) dydt 2

F (t , y, x(t , y))dydt − x (T , ·) , x1 (·)




L2 (Ω )

,

(50)

defined in U. For that problem we assume the following hypotheses: GG1. F (t , y, x) is measurable with respect to (t , y) in (0, T ) × Ω for all x in R, F (t , y, ·), (t , y) in (0, T ) × Ω , and G(·) are continuously differentiable and convex in R. Fx (t , y, x) = Fx1 (x)+ F 2 (t , y), (t , y, x) ∈ (0, T )× Ω × R, F 2 (·, ·) ∈ L1 (0, T ; L2 (Ω )). GG2. There is a ball G with center at zero in U such that Gx (x), Fx1 (x) ∈ L1 (0, T ; L2 (Ω )) and kGx (x)kL1 (0,T ;L2 (Ω )) is



bounded in G. There exists a constant E, such that for some xˆ ∈ G, xˆ U ≤ C Fx1 xˆ L1 (0,T ;L (Ω )) + F 2 (·, ·) L1 (0,T ;L (Ω )) + 2 2






+ E w and Fx1 xˆ L1 (0,T ;L (Ω )) ≤ E. 2 GG3. F (t , y, x) and G(x) satisfy G3.



Gx (ˆx) 1

L (0,T ;L2 (Ω ))






A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

1507

Define the sets: XFG =

X FG

n



x ∈ G ⊂U : kxkU ≤ C Fx1 (x) L1 (0,T ;L (Ω )) + F 2 (·, ·) L1 (0,T ;L (Ω )) , 2 2





 o

+ kGx (x)kL1 (0,T ;L2 (Ω )) + E w , Fx1 (x) L1 (0,T ;L (Ω )) ≤ E , 2 n 

= v ∈ U : kvkU ≤ C Fx1 (v) L1 (0,T ;L (Ω )) + F 2 (·, ·) L1 (0,T ;L (Ω )) 2 2 o  w + kGx (x)kL1 (0,T ;L2 (Ω )) + E for some x ∈ XFG .

(51)

By GG2 X FG is bounded in U. Next define the set X FGd : an element (p, q, z ) ∈ L2 (0, T ; L2 (Ω )) × L2 (0, T ; L2 (Ω )) × L2 (0, T ; L2 (Ω )) belongs to X FGd provided that there exists x ∈ XFG such that for a.e. (t , y) ∈ (0, T ) × Ω pt (T − t , ·) + div q (t , ·) + z (t , ·) = Fx (t , ·, x(t , ·)) q (t , y) = ∇ x (t , y)

with

and z (t , y) = Gx (x(t , y))

or else p (T − t , y) = xt (t , y)

with q (t , y) = ∇ x (t , y) and z (t , y) = Gx (x(t , y)).

The dual functional to (50) is taken then as JDFG

(p, q, z ) = −

T

Z

Z Ω

0 T

Z

Z

+ Ω

0

Z

F ∗ (t , y, (pt (T − t , y) + div q (t , y) + z (t , y))) dydt G∗ (z (t , y))dydt −

1 2

T

Z 0

Z Ω

|q (t , y)|2 dydt +

1

T

Z

2

0

Z Ω

|p (T − t , y)|2 dydt


x¯ Γ (t , y) (q (t , y) , ν(y)) dydt + x (·) , p¯ (T , ·) L2 (Ω ) , 0

− Σ

(52)

where G∗ is the Fenchel conjugate of G and JDFG : L2 (0, T ; L2 (Ω )) × L2 (0, T ; L2 (Ω )) × L2 (0, T ; L2 (Ω )) → R. Lemma 25. Define in XFG the map XFG 3 x → H (x) = v where v is a weak solution to problem

vtt (t , y) − ∆v(t , y) + Fx (t , y, v(t , y)) = Gx (x(t , y)) in (0, T ) × Ω . v(0, y) = x0 (y), vt (0, y) = x1 (y), y ∈ Ω , v(t , y) = x¯ Γ (t , y), (t , y) ∈ Σ .

(53)

Then H (XFG ) ⊂ X FG . Proof. Fix arbitrary x ∈ XFG . Since x ∈ G, by the assumptions on G, see GG2, it follows that Gx (x) ∈ L1 (0, T ; L2 (Ω )). Hence by Theorem 23 there exists a solution v ∈ G of the initial problem for the Eq. (53) satisfying

kvkU ≤ C (kFx (v)kL1 (0,T ;L2 (Ω )) + kGx (x)kL1 (0,T ;L2 (Ω )) ) + E w . Thus the last relation implies that for an arbitrary x ∈ XFG there exists v = H (x) ∈ X FG .



Remark 26. Let us observe that if x¯ is a solution to (49) then by the above lemma it has to belong to XFG . Since by GG1, GG2 the Lemma 14 is true also in the case of (49), thus analogously as in the case of functional J F we prove the following Lemma 27. The functional J FG attains its minimum on XFG in X FG i.e. inf J FG (x) = J FG (x) ,

x∈XFG

where x ∈ X FG . Theorem 28. Let J FG (x) = infx∈XFG J FG (x). Then there exists (¯p, q¯ ) ∈ L2 (0, T ; L2 (Ω )) × L2 (0, T ; L2 (Ω )) such that for a.e. (t , y) ∈ (0, T ) × Ω , p¯ (T − t , y) = x¯ t (t , y),

(54)

q¯ (t , y) = ∇ x¯ (t , y),

(55)

−¯pt (T − t , y) − div q (t , y) − Gx (¯x (t , y)) + Fx (t , y, x¯ (t , y)) = 0

(56)

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A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

and J FG (¯x) = JDFG (¯p, q¯ ) . Moreover by Remark 26 x¯ ∈ XFG . Example 29. We show, how to use, the above theorem to solve the nonconvex superlinear problem: xtt − ∆x + 3x2 = 0, in (0, T ) × Ω , x(t , y) = x¯ Γ (t , y), (t , y) ∈ Σ , x(0, ·) = x0 (·) ∈ H 1 (Ω ), xt (0, ·) = x1 (·) ∈ L2 (Ω ).

(57)

Let us put x3 = x4 + x3 + x2 − x4 − x2 . Then assume F (t , y, x) = x4 + x3 + x2 + x and G(x) = x4 + x2 + x. F (t , y, ·) and G(·) are convex. Assume n = 3 and choose such a T ≤ 1 that C = 2. Ω = B(0, 1) and take such initial values and boundary conditions that E w = 2. Put G = BU (0, 2) (ball at zero with radius 2 in U). Note that for xˆ (·, ·) = 1,

 





Gx xˆ 1

xˆ ≤ C Fx xˆ 1 + Ew. + L (0,T ;L2 (Ω )) L (0,T ;L2 (Ω )) U



Take E = 10, then Fx xˆ 1 < 10. Note that Gx xˆ 1 L (0,T ;L2 (Ω ))

L (0,T ;L2 (Ω ))

≤ 4. Thus assumptions GG1 and GG2 (and of

course GG3) are satisfied, so by the above theorem there exists x ∈ XFG (nontrivial) being a solution to (57). Proof of the Theorem 28. Let x¯ ∈ X FG be such that J FG (¯x) = infx∈XFG J FG (x). We define, for all (t , y) ∈ (0, T ) × Ω ,

− p¯ t (T − t , ·) = div q¯ (t , ·) + z¯ (t , ·) − Fx (t , ·, x¯ (t , ·)) ,

(58)

with q¯ and z¯ given by q¯ (t , y) = ∇ x¯ (t , y), FG

By the definitions of J , J FG (¯x) =

T

Z

Z  T

Z

1 2

Ω

0

z¯ (t , y) = Gx (t , y, x¯ (t , y)) .

JDFG ,

Z

≤− Ω

0

−

T

Z

1 2

Ω

Z

0 T

Ω

Z

+

1

Ω T

Z

2

0

Z − Σ

|¯q (t , y)|2 dydt −

Z

T

Z

0 T

Z Ω

Z Ω

0

1 2

Z

T

Z Ω

0

|¯p (T − t , y)|2 dydt +

Ω

|¯p (T − t , y)|2 dydt +

Ω

2 (Ω )

h∇ x¯ (t , y) , q¯ (t , y)i dydt

F (t , y, x¯ (t , y))dydt

T

Z

Z

h¯x (t , y) , z¯ (t , y)i dydt

F (t , y, (¯pt (T − t , y) + div q¯ (t , y) + z¯ (t , y))) dydt −

Z

T

Z 0

∗

=− 0

2


|¯xt (t , y)|2 − G (¯x (t , y)) + F (t , y, x¯ (t , y)) dydt − x¯ (T , ·) , x1 (·) L

G∗ (¯z (t , y))dydt +

+ Z

1

h¯xt (t , y) , p¯ (T − t , y)i dydt +

0 T

Z

relations (55), (56) and the Fenchel–Young inequality it follows that

|∇ x¯ (t , y)|2 −

Z

(59)

Z

0

1 2

T

Z

Z

0

Ω

|¯q (t , y)|2 dydt

G∗ (¯z (t , y))dydt + x0 (·) , p¯ (T , ·)




Ω

L2 (Ω )

x¯ Γ (t , y) (¯q (t , y) , ν(y)) dydt = JDFG (¯p, q¯ , z¯ ) .

Therefore we get that J FG (¯x) ≤ JDFG (¯p, q¯ , z¯ ) .

(60)

It is clear that the above (p, q, z¯ ) is a limit of certain sequences p , q , z ∈ X FGd . We observe that there are two possible  j j j  forms of sequences p , q , z ⊂ X FGd corresponding to the sequence xj accordingly to the definition of the set X FGd with qj = ∇ xj . Namely, for all (t , y) ∈ (0, T ) × Ω , 

qj (t , y) = ∇ xj (t , y) ,

j

pj (T − t , y) = xt (t , y) ,

j

j

j

z j (t , y) = Gx (t , y, xj (t , y))

(61)

or

−pjt (T − t , ·) = div qj (t , ·) + z j (t , ·) − Fx (t , ·, xj (t , ·)), q (t , y) = ∇ x (t , y) , j

j

(62)

z (t , y) = Gx (x (t , y)). j

j

First we investigate the convergence of both sequences. Since xj converges pointwise to x thus Fx t , y, xj (t , y)



Gx xj (t , y)









and converge pointwise to Fx (t , y, x¯ (t , y)) and {Gx (t , y, x¯ (t , y))}, respectively. As for the sequence (61) we

A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

 j

obviously get, since L2 (0, T ; H j xt

−1

x

n o j

converges strongly in C ([0, T ]; H 1 (Ω )),

1509

weakly in L2 (0, T ; L2 (Ω )) and

xt

n o j

pt

weakly in

(Ω ))

* xt = p ,

∇ xj → ∇ x = q,

z j → z¯ .

By the Fenchel–Young inequality we have J FG (¯x) = inf J FG (x) = lim inf J FG (xj ) j→∞

x∈XFG

T

Z

Z 

≥ lim inf j→∞

2

Ω

0

T

Z

1

Z

T

0





F t , y, xj (t , y) dydt − xj (T , ·) , x1 (·)







+ lim inf j→∞




1 j 2 xt (t , y) − G xj (t , y) dydt 2

|∇ xj (t , y)|2 −

Ω

T

 L2 (Ω )

Z TZ |p (T − t , y)|2 dydt + G∗ (¯z (t , y))dydt 2 0 Ω 2 0 Ω 0 Ω Z TZ Z T D E
j + F (t , y, x(t , y)) dydt + lim inf xj (t , ·) , pt (T − t , ·) + x0 (·) , pj (T , ·) L 1 −1

≥−

1

Z

T

Z

+ Ω

0

=−

1

Z

2

0 T

Z

T

1

|q (t , y)|2 dydt +

T

Z

− Ω

0

2

Z Ω

0

Z

G (¯z (t , y))dydt −

T

Z

1

|¯q (t , y)|2 dydt + ∗

Ω

0

2 (Ω )

H ,H

0

 dt

hx(t , y), (−¯pt (T − t , y) − div q¯ (t , y) − z¯ (t , y))i dydt

Ω

Z

Z

j→∞

Z

+ Z

Z

Ω

0

Z

Z

|p (T − t , y)|2 dydt

x¯ Γ (t , y) (¯q (t , y) , ν(y)) dydt + x0 (·) , p¯ (T , ·)




Σ

L2 (Ω )

F ∗ (t , y, (¯pt (T − t , y) + div q¯ (t , y) + z¯ (t , y))) dydt = JDFG (¯p, q¯ , z¯ ) .

Hence we infer inequality J FG (¯x) ≥ JDFG (¯p, q¯ , z¯ ) and this together with (60) imply equality J FG (¯x) = JDFG (¯p, q¯ , z¯ ). Moreover J FG (¯x) = JDFG (¯p, q¯ , z¯ ), implies T

Z

Z Ω

0

F ∗ (t , y, (¯pt (T − t , y) + div q¯ (t , y) + z¯ (t , y))) dydt + T

Z

Z Ω

0

=

Z Ω

0

+ 1

T

Z

T

Z

2

Z Ω

0

Z − Σ

G∗ (¯z (t , y))dydt +

1

T

Z

2

Z Ω

0

|¯p (T − t , y)| dydt + 2

1

T

Z

2

0

|q (t , y)|2 dydt + Z Ω

1

Z

2

T

F (t , y, x¯ (t , y)) dydt −

Z Ω

0

T

Z

Z

0

Ω

G (¯x (t , y)) dydt

|∇ x¯ (t , y)|2 dydt

|¯xt (t , y)|2 dydt

x¯ Γ (t , y) (¯q (t , y) , ν(y)) dydt + x0 (·) , p¯ (T , ·)




L2 (Ω )

+ x¯ (T , ·) , x1 (·)




L2 (Ω )

.

Therefore by (59), (58) and standard convexity arguments 1 2

T

Z 0

Z Ω

|¯p (T − t , y)|2 dydt +

1

T

Z

2

0

Z Ω

|¯xt (t , y)|2 dydt =

T

Z 0

Z Ω

h¯xt (t , y) , p¯ (T − t , y)i dydt .

Hence by the properties of the Fenchel–Young transform we obtain that p¯ (T − t , y) = x¯ t (t , y) i.e. (54).



4.4. More general case Let us consider now a sequence of convex functions F 1 , F 2 , . . . , F m of the variable x and a function G of the variable (t , y). Let j1 , . . . , jm be a sequence of numbers having values either −1 or +1. Let us assume that our original nonlinearity (see (5)) has the form l = j1 Fx1 + j2 Fx2 + · · · + jm Fxm + G.

(63)

To prove existence of solution for (5) with nonlinearity l just defined we use induction argument. To this effect let us put lm−1 = j1 Fx1 + j2 Fx2 + · · · + jm−1 Fxm−1 and consider the problem xtt (t , y) − ∆x(t , y) + lm−1 (x(t , y)) + G(t , y) = 0, x(0, y) = x (y),

xt (0, y) = x (y),

x(t , y) = x¯ Γ (t , y),

( t , y) ∈ Σ .

0

1

y ∈ Ω,

in (0, T ) × Ω , (64)

1510

A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

For that problem we assume the following hypotheses: Gm−1 1. F 1 , F 2 , . . . , F m−1 are continuously differentiable and convex in R, G (·, ·) ∈ L1 ([0, T ]; L2 (Ω )). Gm−1 2. There exists constant Em−1 , such that for some xˆ ∈ U,

 


w

xˆ ≤ C lm−1 xˆ 1 k + G ·)k (·, 1 L (0,T ;L2 (Ω )) + E L (0,T ;L2 (Ω )) U


and lm−1 xˆ L1 (0,T ;L (Ω )) ≤ Em−1 . 2

Define set

o

n

XGm−1 = x ∈ U : kxkU ≤ C (klm−1 (x)kL1 (0,T ;L2 (Ω )) + kG (·, ·)kL1 (0,T ;L2 (Ω )) ) + E w .

(65)

Induction hypothesis: IH Under hypothesis Gm−1 1, Gm−1 2 problem (64) has a solution in XGm−1 . Consider now our problem (see (5)) with l defined by (63). Put F¯xm (·, ·, x) = jm Fxm (x) + G (·, ·). We assume the following hypotheses: Gm Assume Gm−1 1, Gm−1 2. Let F m be continuously differentiable and

convex

in R. There exists a ball F in U with center at zero, such that Fxi (x) ∈ L1 ([0, T ]; L2 (Ω )), i = 1, . . . , m, for x ∈ F and Fxm (x) L1 ([0,T ];L (Ω )) is bounded in F. For constant Em−1 2

there exists some xˆ ∈ F, such that xˆ U ≤ C ( lm−1 xˆ L1 (0,T ;L (Ω )) + Fxm (ˆx) L1 (0,T ;L (Ω )) + kG (·, ·)kL1 (0,T ;L2 (Ω )) ) + E w and 2 2




lm−1 xˆ 1 ≤ E . m − 1 L (0,T ;L (Ω ))








2

Gm 3 F 1 , F 2 , . . . , F m satisfy G3. Define the sets m XFG =

X mFG

n

x ∈ F : kxkU ≤ C (klm−1 (x)kL1 (0,T ;L2 (Ω )) + Fxm (x) L1 (0,T ;L (Ω )) 2



o + kG (·, ·)kL1 (0,T ;L2 (Ω )) ) + E w , klm−1 (x)kL1 (0,T ;L2 (Ω )) ≤ Em−1 , (66) o n

m . (67) = v ∈ F : kvkU ≤ C (klm−1 (v)kL1 (0,T ;L2 (Ω )) + Fxm (x) L1 (0,T ;L (Ω )) + kG (·, ·)kL1 (0,T ;L2 (Ω )) ) + E w , x ∈ XFG 2

Example 30. Following the same argumentation as in Examples 22 and 29 (with the same Ω ) we get that l being a polynomial in x (taking into account the Sobolev’s embedding H 1 (Ω ) → L 2n (Ω )) satisfies Gm for some suitable chosen n−2

ball F. Remark 31. The last example shows that enough large class of nonlinear functions l can be treated by the method presented in the paper. Similarly as in the F − G case, using now hypothesis IH one can prove the following lemma. m m Lemma 32. Define in XFG the map XFG 3 x → H (x) = v where v is a solution to problem

vtt (t , y) − ∆v(t , y) + lm−1 (v(t , y)) = −F¯xm (t , y, x(t , y)) on (0, T ) × Ω x(0, y) = x0 (y),

xt (0, y) = x1 (y),

x(t , y) = x¯ Γ (t , y),

(t , y) ∈ Σ .

Then H (

m XFG

)⊂X

mFG

y ∈ Ω,

.

Define functional corresponding to problem (5) with l defined by (63): J mFG (x) =

T

Z

Z 

2

Ω

0

T

Z

1

Z

1 2


|xt (t , y)|2 dydt − x (T , ·) , x1 (·) L

2 (Ω )

j1 F 1 (x(t , y)) + j2 F 2 (x(t , y)) + · · · + F¯ m (t , y, x(t , y)) dydt .




+ 0

|∇ x(t , y)|2 −

Ω

(68)

Of course, in our case, J = J mFG , but writing J mFG we would like to stress the form of the last term in J mFG . Next define the set X mFGd : an element (p, q, z1 , . . . , zm−1 ) ∈ L2 (0, T ; L2 (Ω )) × L2 (0, T ; L2 (Ω )) × · · · × L2 (0, T ; L2 (Ω )) belongs to X mFGd m provided that there exists x ∈ XFG such that for a.e. (t , y) ∈ (0, T ) × Ω pt (T − t , ·) + div q (t , ·) + z1 (t , ·) + · · · + zm−1 (t , ·) = j1 Fx1 (x(t , ·)) with q (t , y) = ∇ x (t , y)

and

z1 (t , y) = −j2 Fx2 (x(t , y)), . . . , zm−1 (t , y) = −F¯xm (t , y, x(t , y))

A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

1511

or else p (T − t , y) = xt (t , y) z1 (t , y) = −

j2 Fx2

with q (t , y) = ∇ x (t , y) and

(x(t , y)), . . . , zm−1 (t , y) = −F¯xm (t , y, x(t , y)).

The dual functional to (68) is taken then as JDmFG (p, q, z1 , . . . , zm−1 ) = −j1

T

Z

Ω

0 T

Z

Z Z

− j2

Ω

0

−

1

T

Z

2

Z Ω

0

F 1∗ ((pt (T − t , y) + div q (t , y) + z1 (t , y) + · · · + zm−1 (t , y))) dydt F 2∗ (z1 (t , y))dydt − · · · −

T

Z

Z

0

2

Z


+ x (·) , p (T , ·) L 0

2 (Ω )

T

Z

1

|q (t , y)|2 dydt +

0

Z Ω

Ω

F¯ m∗ (t , y, zm−1 (t , y))dydt

|p (T − t , y)|2 dydt

x¯ Γ (t , y) (q (t , y) , ν(y)) dydt ,

− Σ

(69)

where F i∗ is the Fenchel conjugate of F i with respect to third variable and JDmFG : L2 (0, T , L2 (Ω )) × L2 (0, T , L2 (Ω )) × · · · × L2 (0, T , L2 (Ω )) → R. Analogously as in the case of functional J FG one can prove the following m Lemma 33. The functional J mFG attains its minimum on XFG in X mFG i.e.

inf J mFG (x) = J mFG (x) ,

m x∈XFG

where x ∈ X mFG . Now we are in position to prove Theorem 12 and in consequence the main theorem of the paper. Proof of the Theorem 12. Define for the above x ∈ X mFG m−1

−¯pt (T − t , ·) = div q¯ (t , ·) +

X

z¯i (t , ·) − j1 Fx1 (¯x (t , ·)) ,

i=1

for all t ∈ (0, T ) with q¯ and z¯i , i = 1, . . . , m − 1 given in (17) and (19). By the definitions of J mFG , JDmFG , relations (17), (18) and the Fenchel–Young inequality it follows that J

mFG

(¯x) =

T

Z

Z

1 2

Ω

0

Z

T

Z

0 T

Z Ω

Z Ω

T

Ω T

Z Ω

T

Z

1

Ω T

Z

2

T

2

2

Z

T

Z

Z Ω

0 T

Z

|¯q (t , y)| dydt − 2

Ω

0

1∗

X Ω

T

Z 0

Z *

m−1

Ω

i =1

x¯ (t , y) ,

X

Ω

X

+ z¯i (t , y) dydt

!

m−1

Z

ji F (¯zi−1 (t , y)) + F¯ i∗

m∗

(t , y, z¯m−1 (t , y)) dydt

i=2

p¯ t (T − t , y) + div q¯ (t , y) +

|¯q (t , y)|2 dydt + m−1

Z

|¯p (T − t , y)|2 dydt

X

! z¯i (t , y) dydt

i=1

Ω

− 0

j1 F

Z

0

Z

1

1

m−1

=− −

T

Z 0

Z

0

ji F (¯x (t , y)) dydt i

i=1

j1 F (¯x (t , y))dydt − 1

+ 0

|¯xt (t , y)| +

h∇ x¯ (t , y) , q¯ (t , y)i dydt −

+ Z

X

h¯xt (t , y) , p¯ (T − t , y)i dydt +

Z

0

2

!

m−1 2




≤− Z

1

F¯ m (t , y, x¯ (t , y)) dydt − x¯ (T , ·) , x1 (·)

+

0

|∇ x¯ (t , y)| − 2

i =2

1 2

T

Z 0

Z

|¯p (T − t , y)|2 dydt − Ω !

Z Σ

x¯ Γ (t , y) (¯q (t , y) , ν(y)) dydt

ji F i∗ (¯zi−1 (t , y)) + F¯ m∗ (t , y, z¯m−1 (t , y)) dydt + x0 (·) , p¯ (T , ·)




L2 (Ω )

.

1512

A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

Therefore we get that J mFG (¯x) ≤ JDmFG (¯p, q¯ , z¯1 , . . . , z¯m−1 ) .

(70)

n

We observe that there are two possible sequences

j

j

pj , qj , z1 , . . . , zm−1

o

 ⊂ X FGd corresponding to the sequence xj

accordingly to the definition of the set X mFGd with qj = ∇ xj . Namely, for all (t , y) ∈ (0, T ) × Ω , j = 1, 2, . . . , j

pj (T − t , y) = xt (t , y) ,

qj (t , y) = ∇ xj (t , y) , j z1

(t , y) = −

j2 Fx2

(x (t , y)), . . . , j

j zm−1

(71)

(t , y) = − ¯ (t , y, x (t , y)) Fxm

j

or m−1

−pjt (T − t , y) = div qj (t , y) +

X

j

zi (t , y) − j1 Fx1 (xj (t , y)),

qj (t , y) = ∇ xj (t , y) ,

(72)

i =1 j

j

z1 (t , y) = −j2 Fx2 (xj (t , y)), . . . , zm−1 (t , y) = −F¯xm (t , y, xj (t , y)). First we investigate the convergence of both sequences. Since xj converges pointwise to x therefore Fxi xj (t , y)


,i =
 1, . . . , m − 1 and F¯xm t , y, xj (t , y) converge pointwise to Fxi (¯x (t , y)), i = 1, . . . , m − 1 and F¯xm (t , y, x¯ (t , y)) , n o n o  j j respectively. Since xj converges strongly in C ([0, T ]; H 1 (Ω )), xt weakly in L2 (0, T ; L2 (Ω )) and pt weakly in 

L2 (0, T ; H −1 (Ω )) j

xt * xt = p,

j

∇ xj → ∇ x = q,

zi → z¯i ,

i = 1, . . . , m − 1.

By the Fenchel–Young inequality we have J mFG (¯x) = inf J mFG (x) = lim inf J mFG (xj ) j→∞

x∈XFG

T

Z

Z

j1 F 1 (xj (t , y)) + F¯ m t , y, xj (t , y)





≥ lim inf j→∞

Ω

0

T Z

Z + lim inf j→∞

≥−

1 2

Ω

0 T

Z

X

− Ω




2

2

1 2

Z

T

i =2

Z Ω

0

|p (t , y)|2 dydt

! Z m∗ ¯ ji F (¯zi−1 (t , y)) + F (t , y, z¯m−1 (t , y)) dydt +

T

i∗

0

i=2 T

L2 (Ω )

! −1 2 m X
1 1 j j 2 i j |∇ x (t , y)| − xt (t , y) + ji F x (t , y) dydt

|q (t , y)|2 dydt + m−1

Z

0

Ω

0

T Z

Z

dydt − xj (T , ·) , x1 (·)

Ω

Ω

j1 F 1 (x(t , y)) dydt


+ x0 (·) , pj (T , ·) L (Ω ) dt 2 j→∞ H 1 ,H −1 0 !+ Z TZ * m − 1 X + x(t , y), p¯ t (t , y) − div q¯ (t , y) − z¯i (t , y) dydt Z

T

Z

2

T

Ω

X Ω

0 T

Z − Σ

1 2

Z 0

T

Z Ω

|p (T − t , y)|2 dydt !

ji F i∗ (¯zi−1 (t , y)) + F¯ m∗ (t , y, z¯m−1 (t , y)) dydt

i=2

!

m−1

Z

− 0

|¯q (t , y)|2 dydt + m−1

Z

− Z

i=1

Z

0

Z

E

Ω

0

1

j

xj (t , ·) , pt (T − t , ·)

+ lim inf

=−

D

Z Z

Ω

j1 F

1∗

p¯ t (T − t , y) + div q¯ (t , y) +

X

z¯i (t , y) dydt

i=1

x¯ Γ (t , y) (¯q (t , y) , ν(y)) dydt + x0 (·) , p¯ (T , ·)




L2 (Ω )

= JDmFG (¯p, q¯ , z¯1 , . . . , z¯m−1 ) . Hence we infer inequality J mFG (¯x) ≥ JDmFG (¯p, q¯ , z¯1 , . . . , z¯m−1 ) and this together with (70) imply equality J mFG (¯x) = JDmFG (¯p, q¯ , z¯1 , . . . , z¯m−1 ). Moreover J mFG (¯x) = JDFG (¯p, q¯ , z¯1 , . . . , z¯m−1 ) , similarly as in former cases, implies

A. Nowakowski / Nonlinear Analysis 73 (2010) 1495–1514

1 2

T Z

Z 0

Ω

|¯p (T − t , y)|2 dydt +

1 2

T Z

Z 0

Ω

|¯xt (t , y)|2 dydt =

T Z

Z 0

Ω

1513

h¯xt (t , y) , p¯ (T − t , y)i dydt

m and in consequence we obtain that p¯ (T − t , y) = x¯ t (t , y) i.e. (16). From (16), (17) and (18) we infer that x¯ ∈ XFG .



4.5. Proof of stability — Theorem 11 In this section we some stability results to our nonlinear case i.e. Theorem 11. To this effect let us assume we are  prove given a sequence x0n in H 1 (Γ ) converging to x¯ 0 in H 1 (Γ ). Let {¯xΓ n } be a sequence of solutions to (4) corresponding to xn . Then by Proposition 13 and its proof we know that JΓt (¯xΓ n ) = 0, t ∈ [0, T ], n = 1, 2, . . . and that JΓt (x) ≥ 0 for all x ∈ U t , t ∈ [0, T ] (moreover JΓt is weakly lower semicontinuous in U t , t ∈ [0, T ]). Therefore the sequence {¯xΓ n } is bounded in H 1 (Γ ) and hence it has a subsequence (which we shall denote again by {¯xΓ n }) converging weakly in H 1 (Γ ) to some x¯ Γ ∈ H 1 (Γ ) and JΓt (¯xΓ ) = 0 on Ut , t ∈ [0, T ]. The last implies that x¯ Γ is a solution to (4) corresponding to x¯ 0 . Next let us assume that x0n , x1n given sequences in H 1 (Ω ), L2 (Ω ) respectively, converging to x¯ 0 , x¯ 1 in H 1 (Ω ), L2 (Ω )

 0

and x0n (y) = x¯ Γ n (0, y) on Γ , n = 1, 2, . . .. Of course x0n , x1n are bounded in H 1 (Ω ), L2 (Ω ) respectively. Therefore, estimations for kxn (t , ·)kH 1 (Ω ) and kxnt (t , ·)kL2 (Ω ) , (solutions to linear equations corresponding to x0n , x1n ) can be taken with

 

the same constants E w > 0 and Aw > 0 for all xn , xnt determined by x0n , x1n , n = 1, 2, . . .. Thus we assume hypotheses (HΓ ), Gm , Gm 3 to be considered and satisfied. In consequence, all the assertions of Theorem 12 are true for all n with estimations independent on n. Because, we have nonlinear problem we cannot expect the same type of continuous dependence as in [14] for linear case. We are only able to prove a theorem of the following type. Theorem 34. Let x0n , x1n given sequences in H 1 (Ω ), L2 (Ω ) respectively, converging to x¯ 0 , x¯ 1 in H 1 (Ω ), L2 (Ω ) with x0n (y) =

 

x¯ Γ n (0, y) on Γ , n = 1, 2, . . .. Then there is a subsequence of {xn }- solutions to (1) corresponding to x0n , x1n , which we denote again by {xn } weakly convergent in H 1 ((0, T ) × Ω ) and strongly in L2 ((0, T ) × Ω ) to an element x¯ ∈ U being a solution to (1) corresponding to x¯ 0 , x¯ 1 .

 

m Proof. First note that for each x0n , x1n there exists a solution x¯ n ∈ XFG ⊂ U to (1) determined by x0n and x1n , n = 1, 2, . . .. m Therefore, choosing suitable subsequence, let x¯ ∈ XFG be a weak limit in H 1 ((0, T ) × Ω ) and strong in L2 ((0, T ) × Ω ) of {¯xn } . We know also that for all n, J mFG (¯xn ) = JDmFG (¯pn , q¯ n ) where {¯pn , q¯ n } denote the sequences corresponding to {¯xn } and satisfying

x¯ nt (t , y) = p¯ n (T − t , y) ,

∇ x¯ n (t , y) = q¯ n (t , y) , p¯ nt (T − t , y) + div q¯ n (t , y) − l (t , y, x¯ n (t , y)) = 0. Since {¯xn } is weakly convergent in H 1 ((0, T ) × Ω ) and strongly in L2 (0, T ; L2 (Ω )), x¯ nt * xt = p, ∇ x¯ n → ∇ x = q in L2 , therefore x¯ n converges pointwise to x. Therefore {¯pnt + div q¯ n } converges pointwise to p¯ t + div q¯ . Hence we infer that J mFG (¯x) = lim J mFG (xn ) = lim JDmFG (pn , qn ) = JDmFG (¯p, q¯ ) n→∞

n→∞

and in consequence xt (t , y) = p (T − t , y) ,

∇ x (t , y) = q (t , y) , pt (T − t , y) div q (t , y) − l (t , y, x (t , y)) = 0 and so we get the assertion of the theorem.



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