ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A QUASILINEAR HYPERBOLIC EQUATION WITH NONLINEAR DAMPING

We prove the existence and uniqueness of a global solution of a damped quasilinear hyperbolic equation. Key point to our proof is the use of the Yosida approximation. Furthermore, we apply a method based on a specic integral inequality to prove that the solution decays exponentially to zero when the time t goes to innit y.


Introduction
Let Ω ⊂ R N be a bounded domain with smooth boundary Γ.In this paper we are concerned with the global existence and asymptotic behavior of solutions to the mixed problem (P ) Here f (•) is a C 1 -class function satisfying f (s) ≥ m 0 > 0 for s ≥ 0 with m 0 constant, {Γ 0 , Γ 1 } is a partition of Γ such that Γ0 ∩ Γ1 = ∅, Γ 0 = ∅, Γ 1 = ∅, and g is a continuous increasing odd function such that g (x) ≥ τ > 0.
Physically, the problem (P) occurs in the study of vibrations of flexible structures in a bounded domain.The motivation for incorporating internal EJQTDE, 1998 No. 7, p. 1 material damping in the quasilinear wave equation as in the first equation of (P) arises from the fact that inherent small material damping is present in real materials.Hence from the physical point of view we say that internal structural damping force will appear so long as the system vibrates.
The problem (P) with h = 0, g(x) = δx (δ > 0) and Γ 1 = ∅ was studied by De Brito [3].She has shown the existence and uniqueness of global solutions for sufficiently small initial data by using a Galerkin method.When g(x) = δx and Γ 0 = ∅, Ikehata [4] has shown the existence of global solutions by a Galerkin method, the key point of his proof is to restrict (P) to the range of −∆ on which −∆ is positive definite.In fact the restricted problem can be solved by a Galerkin method exactly as in De Brito [3].When g is nonlinear, Ikehata's approach seems to be very difficult.The author in [1] has been successful in proving the global existence and establishing the precise decay rate of solutions when Γ 1 = ∅, g is nonlinear without any smallness conditions on the initial data and without the assumption g (x) ≥ τ > 0.
Our study is motivated by Ikehata and Okazawa's work [5] where global existence was proved when g(x) = δx (δ > 0) and Dirichlet or Neumann boundary condition by using the Yosida approximation method together with compactness arguments.In our work, the feedback g is nonlinear, and furthermore we study the asymptotic behavior of the global solution when h = 0.
The contents of this paper are as follows.In section 2, we give our main results.In section 3, we establish the existence of global solutions.In section 4, we study the asymptotic behavior of solutions of (P) with h = 0.

Statement of the main theorems
We define the energy of the solution u to problem (P) by the formula (2.1) where f = s 0 f (t) dt and • n denotes the usual norm of L n (Ω).Our main results are Here B 1 , C 1 are positive constants defined by and BC([0, ∞); L 2 (Ω)) denotes the set of all L 2 (Ω)-bounded continuous functions on [0, ∞).

Theorem 2.2
In addition to the conditions in theorem 2.1, we assume that f is nondecreasing, h = 0 and then we have the decay property Before giving the proofs, we recall the: Let A be a nonnegative selfadjoint operator in a Hilbert space H with the norm | • |, A λ its Yosida approximation and (A λ ) Let E : R + → R + be a non-increasing function and assume that there exists a constant T > 0 such that Let F and G be nonnegative continuous functions on [0, T ].If then where C > 0 is a constant.
Lemma 2.5 is a special case of an inequality that can be found in Bihari [2].

Global existence
First, we solve the approximate problem Problem (P λ ) can be easily solved by successive approximation method.Hence problem (P λ ) has a unique local solution We shall see that u λ (t) can be extended to [0, ∞).

Proof
Multiplying both sides of the first equation in (P λ ) by 2u λ , we have After integration on [0, t], we see that It follows from lemma 2.5 that .

Proof
Multiplying the both sides of the first equation in (P λ ) by −2∆ λ u λ (t), we have It follows that By (3.3) we obtain EJQTDE, 1998 No. 7, p. 5 Integrating this inequality on [0, t], we have then we obtain (3.4).
2) is satisfied, then we have

Proof
First we show that we have by definition Set where we set Therefore, (3.12) follows from lemma 3.1 and lemma 3.4.

Proof
Let u λ (t) be a solution of (P λ ) on [0, T λ ).Since u λ (t) and u λ (t) are uniformly bounded in L 2 (Ω), u λ (T λ ) and u λ (T λ ) exist and we can choose them as new initial values.Moreover, since u λ (t) is uniformly bounded, the local Lipschitz continuity of the mapping u → f (|∇ λ u| 2 )∆ λ u is always verified.Therefore, u λ (t) can be extended onto the semi-infinite interval [0, ∞).

Lemma 3.6
There is a subsequence is the set of all L 2 -valued bounded continuous functions on [0, ∞).

Proof
By the fact that u λ 2 is bounded on [0, T λ ) and lemma 3.1, it follows that J 1/2 λ u λ and ∇ λ u λ belong to BC([0, ∞), L 2 (Ω)).By the definition of ∇ λ we have J this implies that for each t > 0, {J , and then relatively compact in L 2 (Ω).As {J Thus, there exist a subsequence {J By (2.9) we conclude that for any T > 0 Let {λ n } and u(•) be as in lemma 3.6.Assume that (2.2) is satisfied.Then u(•) ∈ BC 1 ([0, ∞), L 2 ) and there is a subsequence {µ n } of {λ n } such that for any T > 0 Here (3.15) can be proved in the same way as in the proof of lemma 3.6, in fact we have (3.16) and (3.17) follow from (3.13) and (3.15) respectively.Therfore we have Let u and {λ n } be as in lemma 3.6.Assume that (2.2) is satisfied.Then u ∈ BC([0, ∞); H 1 Γ 0 ) and for any T > 0 and hence

Proof
We have we have Hence

Lemma 3.9
Let u and {µ n } be as in lemma 3.7, then u has a (strong) derivative u ∈ L ∞ (0, ∞; L 2 ) and and hence Proof ¿From (3.12), we note that u is Lipschitz continuous.Therefore u is differentiable a.e. on (0, ∞) with u ∈ L ∞ (0, ∞; L 2 ).It follows from the previous lemma that 2 )∆u.So we see from the Banach-Steinhauss theorem that Passing to the limit h → 0, we obtain w = u a.e. on (0, ∞).

Lemma 3.10
Let u be as in lemma 3.6.Assume that (2.2) is satisfied, then u is the unique solution to problem (P).
The proof follows immediately from the Gronwall's lemma., 1998 No. 7, p. 10 In this section we consider the problem (P 1)

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The energy defined by (2.1) is such that hence the energy is non-increasing.
Multiplying the first equation in (P1) with u and integrating by parts, we obtain We apply the Young inequality to the two terms of the RHS of (4.4), we obtain ).