GLOBAL EXISTENCE AND ENERGY DECAY OF SOLUTIONS TO THE CAUCHY PROBLEM FOR A WAVE EQUATION WITH A WEAKLY NONLINEAR DISSIPATION

where g : R→ R is a continuous nondecreasing function and λ and σ are positive functions. When we have a bounded domain instead of Rn, and for the case g(x) = δx (δ > 0) (without the term λ2(x)u), Ikehata and Suzuki [8] investigated the dynamics, they have shown that for sufficiently small initial data (u0,u1), the trajectory (u(t),u′(t)) tends to (0,0) inH 0 (Ω)×L2(Ω) as t→ +∞. When g(x)= δ|x|m−1x (m≥ 1, λ≡0, σ ≡ 1), Georgiev and Todorova [4] introduced a new method and determined suitable relations between m and p, for which there is global existence or alternatively finite-time blow up. Precisely they showed that the solutions continue to exist globally in time if m ≥ p and blow up in finite time if m < p and the initial energy is sufficiently negative. This result was later generalized to an abstract setting by Levine and Serrin [12] and Levine et al. [11]. In these papers, the authors showed that no solution with negative initial energy can be extended on [0,∞[, if the source term dominates over the damping term (p > m). This generalization allowed them also to apply their result to quasilinear wave equations (see [1, 17]). Quite recently, Ikehata [7] proved that a global solution exists with no relation between p and m by the use of a stable set method due to Sattinger [18]. For the Cauchy problem (1.1) with λ ≡ 1 and σ ≡ 1, when g(x) = δ|x|m−1x (m ≥ 1) Todorova [21] (see [16]) proved that the energy decay rate is E(t)≤ (1 + t)−(2−n(m−1))/(m−1)


Introduction
We consider the Cauchy problem for the nonlinear wave equation with a weak nonlinear dissipation and source terms of the type where g : R → R is a continuous nondecreasing function and λ and σ are positive functions.
When we have a bounded domain instead of R n , and for the case g(x) = δx (δ > 0) (without the term λ 2 (x)u), Ikehata and Suzuki [8] investigated the dynamics, they have shown that for sufficiently small initial data (u 0 ,u 1 ), the trajectory (u(t),u (t)) tends to (0,0) in H 1 0 (Ω) × L 2 (Ω) as t → +∞. When g(x) = δ|x| m−1 x (m ≥ 1, λ ≡ 0, σ ≡ 1), Georgiev and Todorova [4] introduced a new method and determined suitable relations between m and p, for which there is global existence or alternatively finite-time blow up. Precisely they showed that the solutions continue to exist globally in time if m ≥ p and blow up in finite time if m < p and the initial energy is sufficiently negative. This result was later generalized to an abstract setting by Levine and Serrin [12] and Levine et al. [11]. In these papers, the authors showed that no solution with negative initial energy can be extended on [0, ∞[, if the source term dominates over the damping term (p > m). This generalization allowed them also to apply their result to quasilinear wave equations (see [1,17]). Quite recently, Ikehata [7] proved that a global solution exists with no relation between p and m by the use of a stable set method due to Sattinger [18].
Our purpose in this paper is to give a global solvability in the class H 1 and energy decay estimates of the solutions to the Cauchy problem (1.1) for a weak linear perturbation and a weak nonlinear dissipation.
We use a new method recently introduced by Martinez [13] (see also [2]) to study the decay rate of solutions to the wave equation u − ∆ x u + g(u ) = 0 in Ω × R + , where Ω is a bounded domain of R n . This method is based on a new integral inequality that generalizes a result of Haraux [6]. So we proceed with the argument combining the method in [13] with the concept of modified stable set on H 1 (R n ). Here the modified stable set is the extended R n version of Sattinger's stable set.

Preliminaries and main results
λ(x), σ(t), and g satisfy the following hypotheses.
(i) λ(x) is a locally bounded measurable function defined on R n and satisfies where d is a decreasing function such that lim y→∞ d(y) = 0.
(ii) σ : R + → R + is a nonincreasing function of class C 1 on R + . Consider g : R → R a nondecreasing C 0 function and suppose that there exist C i > 0, i = 1,2,3,4, such that where m ≥ 1 and 1 ≤ r ≤ (n + 2)/(n − 2) + . We first state two well-known lemmas, and then we state and prove two other lemmas that will be needed later.
A. Benaissa and S. Mokeddem 937 Lemma 2.3 [10]. Let E : R + → R + be a nonincreasing function and assume that there are two constants p ≥ 1 and A > 0 such that where c and ω are positive constants independent of the initial energy E(0).
Lemma 2.4 [13]. Let E : R + → R + be a nonincreasing function and φ : R + → R + an increasing C 2 function such that Assume that there exist p ≥ 1 and A > 0 such that where c and ω are positive constants independent of the initial energy E(0).
Before stating the global existence theorem and decay property of problem (1.1), we will introduce the notion of the modified stable set. Let (2.14) 938 Cauchy problem for nonlinear wave equation for u ∈ H 1 (R n ). Then we define the modified stable set ᐃ * and ᐃ * * by Next, let J(u) and E(t) be the potential and energy associated with problem (1.1), respectively: (2.16) We get the local existence solution.
When λ ≡ const, we use the following theorem of local existence in the space H 2 × H 1 , and the decay property of the energy E(t) is necessarily required for the local solution to remain in ᐃ * * as t → ∞; this fact of course guarantees the global existence in H 2 × H 1 and by approximation, we obtain global existence in H 1 × L 1 .
Theorem 2.6 [15]. Let (u 0 ,u 1 ) ∈ H 2 × H 1 . Suppose that Proof of Theorem 2.5 (see [15,18]). Since the argument is standard, we only sketch the main idea of the proof. Let (u 0 ,u 1 ) ∈ H 1 × L 2 and u 0 ∈ ᐃ * . Then we have a unique local solution u(t) for some T > 0. Indeed, taking suitable approximate functions f j such that (see [20]) ). Further, we can prove that u j (t) ∈ ᐃ * , 0 < t < T, A. Benaissa and S. Mokeddem 939 for sufficiently large j, and there exists a subsequence of {u j (t)} which converges to a functionũ(t) in certain senses.ũ(t) is, in fact, a weak solution in C([0,T);H 1 (R n )) ∩ C 1 ([0,T);L 2 (R n )) (see [19,20]) and such a solution is unique by Ginibre and Velo [5] and Brenner [3]. We can also construct such a solution which meets moreover the finite propagation property, if we assume that the initial data u 0 (x) and u 1 (x) are of compact support: Applying [9, Appendix 1] of John, then the solution is also of compact support: supp We denote the life span of the solution u(t,x) of the Cauchy problem (1.1) by T max . First we consider the case λ(x) ≡ const (λ(x) ≡ 1 without loss of generality). And construct a stable set in H 1 (R n ). Setting
. Furthermore, the global solution of the Cauchy problem (1.1) has the following energy decay property: (2.38) Remark 2.9. In Theorem 2.7, the global existence and energy decay are independent, but in Theorem 2.8, we need the estimation of the energy decay for a local solution to prove global existence. (2.39) (2) If σ(t) = 1/t θ lnt ln 2 t ··· ln p t, by applying Theorem 2.7, we obtain For example, if n(m − 1)/2 + θ = 1, that is, 1 < m < 1 + 2/n, Cauchy problem for nonlinear wave equation (3) If σ(t) = 1/t θ and d(r) = 1/r γ with θ ≥ γ by applying Theorem 2.8, we obtain (2.42) In order to show the global existence, it suffices to obtain the a priori estimates for E(t) and u(t) 2 in the interval of existence.
To prove Theorem 2.7 we first have the following energy identity to problem (1.1).
Next we state several facts about the modified stable set ᐃ * .
Lemma 2.12. Suppose that (2.44) Proof of Lemma 2.13. Suppose that there exists a number t * ∈ [0,T max [ such that u(t) ∈ ᐃ * on [0, t * [ and u(t * ) ∈ ᐃ * . Then we have it follows from the nonincreasing of the energy that for all t ∈ [0,t * ], where C 0 is the constant defined by (2.22). Note that from (2.55) and our hypothesis Hence, from Lemma 2.11 and (2.60) we get for some M 2 > 0. The above inequality and the continuation principle lead to the global existence of the solution, that is, T max = ∞.
Proof of the energy decay. From now on, we denote by c various positive constants which may be different at different occurrences. We multiply the first equation of (1.1) by E q φ u, where φ is a function satisfying all the hypotheses of Lemma 2.4. We obtain (2.63) A. Benaissa and S. Mokeddem 945 we deduce that for every ε > 0. Also, applying Hölder's and Young's inequalities, we have 946 Cauchy problem for nonlinear wave equation for every ε > 0. Choosing ε and ε small enough, we obtain (2.67) Since xg(x) ≥ 0 for all x ∈ R, it follows that the energy is nonincreasing, locally absolutely continuous and Proof of (2.26). We consider the case m = 1, that is, Then we have where ρ(t,s)=σ(t)g(s) for all s ∈ R. Therefore we deduce from (2.67) (applied with q = 0) that It is clear that φ is a nondecreasing function of class C 2 on R + . The hypothesis (2.23) ensures that Then we deduce from (2.70) that For t ≥ 0, consider First we note that for every t ≥ 0, Next we deduce from (2.2) and (2.3) that for every t ≥ 0, (i) if x ∈ Ω 1 , then u 2 ≤ ((1/σ(t))u ρ(t,u )) 2/(m+1) , (ii) if x ∈ Ω 2 , then u 2 ≤ (1/σ(t))u ρ(t,u ). Hence, using Hölder's inequality, we get that (2.78) Set ε > 0; thanks to Young's inequality and to our definitions of p and φ, we obtain (2.79) We choose φ such that and thanks to Lemma 2.4 (applied with c = 0) we obtain (2.83) Proof of Theorem 2.8. First, we see that if u ∈ ᐃ * * , then In the proof, we often use the following inequality: Now, we assume that I(u 0 ) > (1/2) ∇ x u 0 2 2 . Then for some interval near t = 0. As long as (2.86) holds, we have J(t) ≡ I(u(t)). Thus (2.87) If σ(t) = ᏻ(d(t)), that is, σ(t) → 0 as t → ∞ more rapidly thand(t), we find the same results of asymptotic behaviour as in Theorem 2.7.