Global solutions for a nonlinear wave equation with the pLaplacian operator, Electron

We study the existence and asymptotic behaviour of the global solutions of the nonlinear equation utt pu + ( ) ut + g(u) = f where 0 < 1 and g does not satisfy the sign condition g(u)u 0.


Introduction
The study of global existence and asymptotic behaviour for initial-boundary value problems involving nonlinear operators of the type goes back to Greenberg, MacCamy & Mizel [3], where they considered the one-dimensional case with smooth data.Later, several papers have appeared in that direction, and some of its important results can be found in, for example, Ang & Pham Ngoc Dinh [1], Biazutti [2], Nakao [8], Webb [10] and Yamada [11].In all of the above cited papers, the damping term −∆u t played an essential role in order to obtain global solutions.Our objective is to study this kind of equations under a weaker damping given by (−∆) α u t with 0 < α ≤ 1.This approach was early considered by Medeiros and Milla Miranda [6] to Kirchhoff equations.We also consider the presence of a forcing term g(x, u) that does not satisfy the sign condition g(x, u)u ≥ 0. Our study is based on the pseudo Laplacian operator which is used as a model for several monotone hemicontinuous operators.
Let Ω be a bounded domain in R n with smooth boundary ∂Ω.We consider the nonlinear initial-boundary value problem where 0 < α ≤ 1 and p ≥ 2. We prove that, depending on the growth of g, problem (1.1) has a global weak solution without assuming small initial data.In addition, we show the exponential decay of solutions when p = 2 and algebraic decay when p > 2. The global solutions are constructed by means of the Galerkin approximations and the asymptotic behaviour is obtained by using a difference inequality due to M. Nakao [7].Here we only use standard notations.We often write u(t) instead u(t, x) and u (t) instead u t (t, x).The norm in L p (Ω) is denoted by • p and in W 1,p 0 (Ω) we use the norm For the reader's convenience, we recall some of the basic properties of the operators used here.The degenerate operator −∆ p is bounded, monotone and hemicontinuous from W 1,p 0 (Ω) to W −1,q (Ω), where p −1 + q −1 = 1.The powers for the Laplacian operator is defined by

Existence of Global Solutions
Let g : Ω × R → R be a continuous function satisfying the growth condition where a, b are positive constants, 1 Theorem 2.1 Let us assume condition (2.1) with σ < p. Then given where p −1 + q −1 = 1.
Next we consider an existence result when σ ≥ p.In this case, the global solution is obtained with small initial data.For each m ∈ N we put where C k > 0 is the Sobolev constant for the inequality u k ≤ C k u 1,p , when W 1,p 0 (Ω) → L k (Ω).We also define the polynomial Q by where is its unique local maximum.We will assume that EJQTDE, 1999 No. 11, p. 3 and where γ = lim m→∞ γ m .
Theorem 2.2 Suppose that condition (2.1) holds with σ > p. Suppose in addition that inital data satisfy (2.6) and (2.7).Then there exists a function Proof of Theorem 2.1: Let r be an integer for which H r 0 (Ω) → W 1,p 0 (Ω) is continuous.Then the eigenfunctions of −∆ r w j = α j w j in H r 0 (Ω) yields a "Galerkin" basis for both H r 0 (Ω) and L 2 (Ω).For each m ∈ N, let us put with the initial conditions u m (0) = u 0m and u m (0) = u 1m , where u 0m and u 1m are chosen in V m so that (2.9) ) is a system of ODEs in the variable t and has a local solution u m (t) in a interval [0, t m ).In the next step we obtain the a priori estimates for the solution u m (t) so that it can be extended to the whole interval [0, T ].
A Priori Estimates: We replace v by u m (t) in the approximate equation (2.8) and after integration we have where G(x, u) = u 0 g(x, s) ds.Now from growth condition (2.1) and the Sobolev embedding, we have that But since p > σ, there exists a constant C > 0 such that and then we have Using the convergence (2.9) and the Gronwall's lemma, there exists a constant C > 0 independent of t, m such that With this estimate we can extend the approximate solutions u m (t) to the interval [0, T ] and we have that Now we are going to obtain an estimate for (u m ).Since our Galerkin basis was taken in the Hilbert space H r (Ω) ⊂ W 1,p 0 (Ω), we can use the standard projection arguments as described in Lions [4].Then from the approximate equation and the estimates (2.12)-(2.15)we get (2.16) EJQTDE, 1999 No. 11, p. 5 Passage to the Limit: From (2.12)-(2.14),going to a subsequence if necessary, there exists u such that u m u weakly star in L ∞ (0, T ; W 1,p 0 (Ω)), (2.17) and in view of (2.15) there exists χ such that (2.20) By applying the Lions-Aubin compactness lemma we get from (2.12)-(2.13) Using the growth condition (2.1) and (2.21) we see that is bounded and g(x, u m ) → g(x, u) a.e. in (0, T ) × Ω.

.23)
With these convergence we can pass to the limit in the approximate equation and then The remainder of the proof follows as before.We apply an argument made by L. Tartar [9].From (2.10) we have where λ 1 is the first eigenvalue of −∆ in H 1 0 (Ω), this implies that We claim that there exists an integer N such that Suppose the claim is proved.Then Q( u m (t) 1,p ) ≥ 0 and from (2.7) and (2.26), u m (t) 2 is bounded and consequently (2.11) follows.
Proof of the Claim: Suppose (2.27) false.Then for each m > N , there exists t ∈ [0, t m ) such that u m (t) 1,p ≥ z 0 .We note that from (2.6) and (2.9) there exists N 0 such that Then by continuity there exists a first t * m ∈ (0, t m ) such that . Now from (2.7) and (2.26), there exist N > N 0 and β ∈ (0, z 0 ) such that EJQTDE, 1999 No. 11, p. 7 Then the monotonicity of Q in [0, z 0 ] implies that and in particular, u m (t * m ) 1,p < z 0 , which is a contradiction to (2.28).
Remarks: From the above proof we have the following immediate conclusion: The smallness of initial data can be dropped if either condition (2.1) holds with σ = p and coefficient a is sufficiently small, or σ > p and the sign condition g(x, u)u ≥ 0 is satisfied.

Asymptotic Behaviour
Theorem 3.1 Let u be a solution of Problem (1.1) given by (a) either Theorem 2.1 with the additional assumption: there exists ρ > 0 such that g(x, u)u ≥ ρG(x, u) ≥ 0, ( Then there exists positive constants C and θ such that The proof of Theorem 3.1 is based on the following difference inequality of M. Nakao [7]. Lemma 3.1 (Nakao) Let φ : R + → R be a bounded nonnegative function for which there exist constants β > 0 and γ ≥ 0 such that Then (i) If γ = 0, there exist positive constants C and θ such that (ii) If γ > 0, there exists a positive constant C such that EJQTDE, 1999 No. 11, p. 8 Let us define the approximate energy of the system (1.1) by Then the following two lemmas hold.
Lemma 3.2 There exists a constant k > 0 such that Proof.In the case (a), condition (3.3) is a direct consequence of (3.1).In the case (b) the result follows from (2.27).In fact, from assumption (2.1) with b = 0 we have Then given δ > 0, Proof.In the case (a), the Lemma is a consequence of (3.1).In the case (b), we use again the smallness of the approximate solutions.
Proof of Theorem 3.1: We first obtain uniform estimates for the approximate energy (3.2).Fix an arbitrary t > 0, we get from the approximate problem (2.8) with f = 0 and v = u m (t) Integrating (3.4) from t to t + 1 and putting we have in view of (2.25)  (3.9) Finally we pass to the limit (3.8) and (3.9) and the proof is complete in view of Lemma 3.3.
Proof of Theorem 2.2: We only show how to obtain the estimate (2.11).