ON THE STRONGLY DAMPED WAVE EQUATION AND THE HEAT EQUATION WITH MIXED BOUNDARY CONDITIONS

Here and T are positive constants, ρ : R → R is a nonincreasing and bounded function, g, h ∈ L1(0,T ;L2(0, )), and G is an operator from a subspace of H 1 into L2. In the case where ρ is not continuous, we will understand ρ(x0), at a point of discontinuity x0, as being the whole interval [ρ(x0 +0), ρ(x0 −0)]. In this case ρ will be a multi-valued function, and the “equal signs” in the last equations of (1.1) and (1.2) will be changed to “belong signs.” So, the boundary conditions at x = will be written, respectively, as


Introduction
In this paper, we study existence of strong solutions and existence of global compact attractors for the following one-dimensional problems.
The existence of global solutions for these two problems can be obtained using the theory of monotone operators.The problem (1.2) gives rise to a maximal monotone operator A that is of subdifferential type, A = ∂ϕ, where ϕ is a lower semicontinuous and convex functional.This problem was studied in [1] under some conditions on G, in particular the existence of strong solutions was proved.
Our goal is to obtain existence of global compact attractor.To reach this goal, first of all, we will obtain a relation between the solutions of the two problems.With this relation we can use one problem to get the properties of the other, in particular this relation will be used to prove the existence of strong solutions for the problem (1.1).Once we have existence of solutions, we will start working in order to get the existence of the attractors.For our purpose, we will study the problem (1.2) in two different spaces L 2 and H 1 and using the relation between the solutions we will prove the existence of attractors for the problems.More specifically, setting u t = v, where u(t) is solution operator given by (1.1), we will study the evolution of three operators, z(t) given by (1.2), in the spaces L 2 and H 1 , u(t) + v(t) in the space H 1 and v(t) in the space L 2 .
In order to obtain the results we will use the following procedures: to prove the bounded dissipativeness of the problem (1.1) we will construct an appropriate equivalent norm in the space.The bounded dissipativeness of (1.2) in H 1 will be obtained using the uniform Gronwall lemma with some appropriate estimates.The proof of the compactness of the operators will be done using arguments of Aubin-Lion's type.
Asymptotic behavior of parabolic equations with monotone principal part was recently studied by Carvalho and Gentile in [3], the main difference with our case, problem (1.2), is that our functional ϕ is not equivalent to the norm of the space.

Abstract formulation and existence of solutions
As usual in wave equations context, setting v = u t , (1.1) can be seen as a system: Therefore, our problem (1.1) can be viewed as an evolution equation in the Hilbert space with the inner product Aloisio F. Neves 177 where is given by on the domain Throughout the paper we denote by •, • and | • | the usual inner product and norm of L 2 , respectively.We use the terminology of Brézis [2] and Hale [4] Lemma 2.1.The operator A is maximal monotone.
Proof.If w 1 = (u 1 , v 1 ) and w 2 = (u 2 , v 2 ) are in Ᏸ(A), we have by integrating by parts that Since ρ is nonincreasing and (u i + v i ) ( ) ∈ ρ(v i ( )), i = 1, 2, we have therefore, A is a monotone operator.We prove that A is maximal by showing that R(I + A) = Ᏼ.In fact, if (f, g) ∈ Ᏼ we consider z as being the unique solution of the ODE problem: where a is chosen conveniently.Since z ∈ H 2 (0, ) ∩ H 1,0 and f ∈ H 1,0 , setting Therefore, it remains to be proved that (u + v) ( ) ∈ ρ(v( )) or equivalently z ( ) ∈ ρ(z( )), where We obtain that condition by choosing the constant a appropriately.Setting we have from the variation constant formula we have that the right-hand side of (2.15) is a straight line in plane, parametrized by a, with positive slope.Therefore, there will be a unique a that gives the intersection with the nonincreasing graph of ρ.The lemma is proved.
In order to prove that this weak solution is in fact strong, we will look for a relation between the solutions of (2.1) and the solutions of (1.2).
The problem (1.2) was studied in [1], where G is an operator Aloisio F. Neves 179 not necessarily local and h ∈ L 2 (0, T ; L 2 (0, )).The problem can be written as the abstract evolution problem in L 2 (0, ) where on the domain From Lemmas 2.1 and 2.2 of [1] we have that the operator Ꮽ is strongly monotone, that is, and of subdifferential type, Ꮽ = ∂ϕ, where ϕ : L 2 (0, ) → R ∪ {+∞} is a proper, convex, and lower semicontinuous function defined by where p is given by We should observe that ϕ may assume negative values, but the following estimate is true: where k 1 , k 2 are constants; in particular ϕ is bounded below.Indeed, since |ρ(s)| is bounded (by a constant k), we have for When G is Lipschitz continuous and h ∈ L 2 (0, T ; L 2 (0, )), it was proved, [1, Theorems 3.2 and 4.1], that the solutions of (1.2) are strong, in particular z(t) ∈ Ᏸ(Ꮽ), for every t ∈ (0, T ).Moreover, from Theorem 3.6 of [2] the solution z satisfies Consider the following relations between the problems (2.1) and (1.2): where ξ : [0, ] → R is a smooth function satisfying ξ(0) = 0, ξ( ) = 1, and ξ ( ) = 0.The operator G, given in (2.32), can be considered as an operator from H 1 (0, ) with values in L 2 (0, ), and also with values in H 1,0 .In both of these cases G is Lipschitz continuous and satisfies It is easy to see that if (u, v) is a solution of (2.1) then z, given by (2.31), is a solution of (1.2) with h given by (2.33) and with initial condition z(0) = u(0) + Sv(0).
Conversely, if z is a solution of (1.2), we consider the problem in H 1,0 given by where Since J (t) : H 1,0 → H 1,0 , for t > 0, is globally Lipschitz, this problem has existence and uniqueness of solutions, see [2,Theorem 1.4].If u(t) is this unique solution, then considering v(t) given by the relation (2.31) and g by the relation (2.33) we have that (u, v) satisfies the problem (2.1) with u(0) = 0 and v(0) = z(0).
Under these conditions we can prove the following result.Although we are interested in studying the influence of the nonlinear boundary condition in the problems, we should observe that we have existence of strong solution in more general situations.In fact, we can consider where (q 1 ) the application (t, x) → q(t, x, w) belongs to L 2 (0, T ; L 2 (0, )), for every fixed w ∈ Ᏼ; (q 2 ) there exists k > 0, such that This problem can be viewed as an abstract evolution equation in the Hilbert space Ᏼ ẇ + Aw + B(t, w) = 0, (2.42) where B : [0, T ] × Ᏼ → Ᏼ is given by B = (0, q).(2.43) From the assumptions (q 1 ) and (q 2 ), we have that B satisfies (B 1 ) for every w ∈ Ᏼ, the application t → B(t, w) belongs to L 2 (0, T ; Ᏼ); (B 2 ) there exists k > 0, such that (2.44) Under the above assumptions we have the following result.
The proof is complete.
It is not difficult to see that the strong solutions, given by this theorem, depend continuously on the initial data.More specifically, we have that there exists a positive constant c such that where w(t) and w(t) are solutions of (2.42) with initial conditions w 0 and w0 , respectively.

Existence of attractors in L 2
We start by constructing an equivalent norm in the space Ᏼ.
Lemma 3.1.If W (w) is given by where Moreover, there exists a positive constant λ such that then W 1/2 is an equivalent norm in Ᏼ.
The second part of the lemma follows by noticing that 0 and, for β satisfying (3.2), the right-hand side of this inequality is a negative definite form.), we obtain after an integration by parts that for almost every t 184 Strongly damped wave and heat equations The first integral, line (3.11), can be estimated using Lemma 3.1 To estimate the terms in line (3.12),we observe that (u + v) ( ) satisfies the boundary condition, so it is bounded by some constant M, then using (2.35) we can show that there exists a positive constant c, such that, for every δ > 0 2βu( Using Poincaré inequality we also obtain 0 Choosing δ sufficiently small, we obtain positive constants µ i = 1, 2, and Solving this differential inequality, we obtain that implies (3.8).
In order to prove inequality (3.10) we have that Ꮽ is the subdifferential of the functional ϕ and ϕ(0) = 0, therefore ϕ(z) ≤ Ꮽz, z .So, multiplying (1.2) by z we obtain 1 2 The operator G satisfies (2.34), then, using (2.26), we obtain for every δ > 0 a constant M depending on δ such that and, since we obtain by grouping the equivalent terms and choosing a convenient small value for for some positive constants a 1 , a 2 .Integrating this inequality from t to t +r we obtain This inequality and (3.9) imply (3.10).
Proof.Multiplying (1.2) by φ ∈ H 1,0 , we obtain therefore (3.10) and (3.23) imply that z t ∈ L 2 (0, T ; H 1,0 ) and To prove the compactness it is enough to consider initial data in a dense subset of L 2 (0, ).Let B be the bounded set B = B(r)∩H 1,0 , where B(r) the ball of L 2 (0, ) with center at zero and radius r, and T h (t)(z 0 ) the solution of (1.2) with initial condition z 0 .
Denoting by v u 0 (t) the dynamical system given by the problem (2.1), when the initial condition u(0) = u 0 ∈ H 1,0 is fixed.Using Theorems 3.2 and 3.3 and the relation (2.31), we can state the next result that is a consequence of Theorem 2.2 of Ladyzhenskaya [5].
Theorem 3.4.Under the above conditions the two dynamical systems z(t) and v u 0 (t) have compact global attractors in L 2 (0, ).

Existence of attractors in H 1,0
We start doing some estimates of the solution z(t) of (1.2) when the initial condition z(0) ∈ H 1,0 .Using Theorem 3.6 of [2] for some constants c 1 , c 2 .Thus, from Gronwall inequality, there exists a constant C(ϕ(z(0)), T ) depending on ϕ(z(0)) and T such that ϕ z(t) ≤ C ϕ z(0) , T , (4.5) in particular, we have z ∈ L ∞ (0, T ; H 1,0 ) ∩ L 2 (0, T ; H 2 (0, )).Moreover, if z 1 (t) and z 2 (t) are solutions with initial condition on H 1,0 we have, using (2.23), Since G is Lipschitz, we obtain after an integration on t Aloisio F. Neves 187 therefore, from Gronwall inequality, there exists a constant C depending on T , such that  First of all, we claim that the weak convergence implies the convergence of (z(t n , )).In fact considering a smooth function φ such that φ(0) = 0 and φ( ) = 0, we obtain by integrating by parts (4.12) Thus, passing to the limit, z(t n , ) → z(t, ), what proves our claim.Next, since p is continuous and we have z(t n ) H 1,0 → z(t) H 1,0 that implies the strong convergence of (z(t n )) to z(t) and the continuity of the operator in the variable t.
Now we prove the continuity of the operator in the second variable.In fact, what we have is a stronger result: Theorem 4.2.If (z 0 n ) is a bounded sequence in H 1,0 and converges to z 0 in the L 2 (0, )-norm, then the corresponding solutions of (1.2) z n (t) = T (t)z 0 n converges to z(t) = T (t)z 0 in H 1,0 , for fixed t > 0, as n → ∞.In particular, for t > 0, the operator T (t) : H 1,0 → H 1,0 is compact.
For solution with initial conditions in H 1,0 and bounded in L 2 (0, ), (3.10) implies that t+r t ϕ(z(s))ds is less than a fixed constant for t sufficiently large, then we can use the uniform Gronwall lemma, see [9, page 89], to obtain the result of the theorem.
As a consequence of the two previous theorems and the relation (2.31) we have the following theorem.