EXPONENTIAL BOUNDARY STABILIZATION FOR NONLINEAR WAVE EQUATIONS WITH LOCALIZED DAMPING AND NONLINEAR BOUNDARY CONDITION

. Let D ⊂ R d be a bounded domain in the d − dimensional Euclidian space R d with smooth boundary Γ = ∂D. In this paper we consider exponential boundary stabilization for weak solutions to the wave equation with nonlinear boundary condition: 1 , where (cid:107) u 0 (cid:107) < λ β , E (0) < d β , where λ β , d β are deﬁned in (21), (22) and Γ = Γ 0 ∪ Γ 1 ¯Γ 1

1. Introduction. The nonlinear wave equations with damping or and source terms are well investigated by many authors and there are many results in the literatures(e.g. see Temam [14] and Malek et al. [11], and references therein). Let D ⊂ R d be a bounded domain in the d−dimensional Euclidian space R d with smooth boundary Γ. The local or global existence of the weak solutions to the wave equation u tt (t) − ∆u(t) + β |u t (t)| q u t (t) = α |u(t)| q u(t), α, β ≥ 0.
with an initial value (u 0 , u 1 ) and the Dirichlet condition have been extensively investigated.
On the other hand the wave equations with the nonlinear boundary condition are considered by many authors ( [1,2,5,7,8,9,10,15,17] and references therein). We are concerned with the stability of the weak solution to the wave equations with the source term. Thus it is seemed that it is worth investigating the stability of the weak solutions by the boundary stabilization for the wave equations with the source term. We refer [2] as the investigations of this area.
The exponential stability of solutions to the damped wave equations without the source term but with nonlinear boundary condition is well considered by many authors. In [7] S. Gerbi and B. Said-Houari investigated local existence and exponential growth of solutions for a semilinear damping wave equation with nonlinear boundary conditions. In [17] Zhang and Miao considered the existence and exponential decay of the energy of weak solutions to some damping wave equation with nonlinear boundary conditions. In [16], E. Vitillaro discussed global existence for the wave equation with nonlinear boundary damping and source terms. See also [1,5,8,10] and the references therein.
In [2] M.M. Cavalcanti, V.N.D. Cavalcanti and P. Martinez investigated the existence, uniqueness and the uniform decay of the energy of weak solutions to the weakly damped wave equation with the source term f (s) = |s| q s and the nonlinear boundary condition. We will extend the some results in [2] by investigating exponential boundary stabilization of the weak solutions to (1).
The contents of this paper are as follows. In Section 2 we give Preliminaries. In Sections 3 using the the Galerkin method we prove existence and uniqueness of local weak solutions in time to (1). In Section 4 we consider the global existence of weak solutions to (1) and in Section 5 exponential decay of the energy E(t) of the weak solution u(t) to (1) is investigated.
In this paper the prime denotes the derivative with respect to time t. The c and c * denote positive constants depending only on q and the domain D which change from line to line.

2.
Preliminaries. Throughout this paper we use is the normal outward vector on x. Let H 1 Γ0 (D) be Hilbert space with scalar product and norm: We use also the notations We use the next known lemma: , then there exists a constant c * = C(D, k) such that |u| k ≤ c k u for any u ∈ H 1 0 (D).
We often use the c * instead of the constant c k . The proof of the following lemma is given in Ono [13].  Finally we give the definition of a weak solution to (1).
. Then (u, u t ) is called a weak solution to (1) with an initial value (u 0 , u 1 ) if (1) the following equality holds: for any φ ∈ H 1 0 (D) We need the following conditions in this paper.
3. Existence of local weak solutions. In this section we consider the existence of the local weak solution (u, u t ) to the equation (1) for an initial value (u 0 , u 1 ) ∈ H 1 Γ0 (D) × L 2 (D) by using the Galerkin method. Let {e 1 , e 2 , e 3 , . . . , e n , . . .} be the special orthonormal basis of H 1 Γ0 (D) ∩ H 2 (D) taken as in the proof of Proposition 3.1(p.127, [2]). Set where ϕ nk (t) are the local solutions of the next ordinary differential equation k = 1, 2, 3, . . . with an initial value u n (0) = n k=1 (u 0 , e k )e k and u n t (0) = n k=1 (u 1 , e k )e k , where u t (t) := d dt u(t) and u tt (t) := d dt u t (t). Then it holds that (u n tt (t), w) + ρ(t)(∇u n (t), ∇w) + ρ(t)(γ(u n t (t)), w) Γ1 By the Green theorem We have with the compatible condition ∂u0 ∂ν + γ(u 1 ) = 0 on Γ 1 . Assume that Conditions 1-6 are satisfied. Then there exist a time T 0 > 0 and a unique local weak solution (u, u t ) in time to (1) with the initial value , ∆u n t (t)) and hence by Condition 3 Thus using the Sobolev lemma, the Hölder inequality, the Young inequality and Conditions 2 and 4-6, we have a constant c > 0 such that This means Thus from (7) we obtain positive constants c 7 > 0 and c 8 > 0 such that Since we assume that (u 0 , , we obtain a time T ∈ (0, 1), positive constants c > 0 and B γ > 0 such that Therefore, by the modified Gronwall inequality(Lemma 4.1) there exist a constant M c > 0 uniform in n > 0 and a time 0 < T 0 < T such that And hence by (8) and (10) there exists a constantM c > 0 such that Thus there exist a subsequence {l k } of {n} and a χ ∈ L 2 (0, T 0 ; L 2 (Γ 1 )) such that Therefore by the Aubin-Lions compactness lemma there exists a subsequence {α k } of {l k } such that We also use the same notation {n} instead of {α k } . Then from (5) and (12)  Thus we obtain that Since γ is a monotone nondecreasing function(by Condition 2), for any Ψ ∈ L 2 (0, T 0 ; L 2 (Γ 1 )) t 0 (ρ(s)γ(u n t (s)) − ρ(s)γ(Ψ(s)), u n t (s) − Ψ(s)) Γ1 ds ≥ 0 and hence taking lim inf as n → ∞, it holds that
Thus by the Green theorem Thus by the generalized Hölder inequality Then by the Sobolev lemma, there exists a constant c * > 0 such that u k (t) qd ≤ c * ∆u k (t) 2 . Using (10) we obtain a constant M d > 0 such that By the Gronwall lemma, we have a weak solution (u, u t ) ∈ H 1 Γ0 (D) × L 2 (D) to (1) with the initial value (u 0 , u 1 ). By the usual method we can prove the uniqueness of the weak solutions. This completes the proof of the theorem.

Existence of global weak solutions.
For the function f with Condition 5, furthermore we add the following condition: Condition 8. Let ρ 0 be a positive constant. The function ρ satisfies that In this section we consider the energy E(t) of the weak solution to (1), where E(t) is defined by If 0 < q ≤ 4 d−2 , then by the Poincaré lemma we define the constant K α as Here we define the constant L β by Consider the function Φ(λ) = 1 2 λ 2 ρ 0 − L β λ q+2 , λ ≥ 0. And hence Φ (λ) = λρ 0 − (q + 2)L β λ q+1 . Then there exists the positive constant λ β such that Φ (λ β ) = 0 and it holds that We define the constant d β by It holds that d β > Φ(λ) for all λ ∈ (0, λ β ). By using Conditions 3, 4 7 and 8, we have that Next if there exists a time t 0 > 0 such that u(t 0 ) = λ β and u(t) < λ β for all t ∈ [0, t 0 ), then by (22) and Lemma 4.1 it follows that which means a contradiction. Thus we have the following lemma (See Lemma 3.2, [2]). See also [15]. Furthermore, we have the following lemma (see [2], p124, (2.12)).

Lemma 4.3.
Assume that u 0 < λ β and E(0) < d β . Assume that Conditions 3, 4 7 and 8 are satisfied. Then it holds that Proof. By using Lemma 4.2, we have that .
This completes the proof of the lemma.
We consider the existence of global weak solutions.
We prove the main theorem in this paper. The real number c q+2 in this theorem denotes the Poincaré number, that is, |u| q+2 ≤ c q+2 u holds by Lemma 2.1.