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We consider a wave equation in a bounded domain with nonlinear dissipation and nonlinear source term. Characterizations with respect to qualitative properties of the solution: globality, boundedness, blow-up, convergence up to a subsequence towards the equilibria and exponential stability are given in this article.


Introduction
Let Ω ⊆ R N (N ≥ 1) be a bounded domain with smooth boundary ∂Ω.We are concerned with the behavior of the following superlinear wave equation with dissipation where ω ≥ 0, µ ≥ 0, m ≥ 1, p > 1, and for ω > 0 (1.2) We introduce some related works first and then explain in detail which are our main results.For the well posedness of problem (1.1) and why the natural regularity for the initial data is precisely that of (1.3), we refer to [8].Equations with damping terms have been considered by many authors.For equations with linear weak damping, we refer to [7,10,14].For equations with possibly nonlinear weak damping, we refer to [9,12,16,20,23].Much less work is known for equations with strong damping, see the seminal paper by Levine [15] and also [18,19], but still many problems unsolved.
Gazzola et.al.[8] discussed the case when the weak damping term and the strong damping term are both linear (m = 1 in (1.1)).It is our purpose to shed some further light on damped wave equations of the kind in the problem (1.1) in both presence of nonlinear weak damping and linear strong damping.
Cazenave [5] proved the boundedness of global solutions to (1.1) for ω = µ = 0, while Esquivel-Avila [7] recovered the same result for ω = 0 and µ > 0 and showed that this property may fail in presence of nonlinear disspation, however, by exploiting the same technique in [7], we proved, under the restrictions E(t) ≥ d, ∀ t ≥ 0 (the energy goes beyond the mountain pass level all the times) and m < p, the global solutions can still be bounded even in presence of nonlinear weak damping.
From a different angle of consideration, it is interested to find out for which initial data (1.3) problem (1.1) does have a global solution.For the weakly damped case(ω = 0, µ > 0), Iketa [12] proved that the solution is global and converges to equilibria φ ≡ 0 as t → ∞ if and only if E(0) < d and u 0 ∈ N + .In Theorem 4.2 we extend this result to the case ω > 0. For related asymptotic stability results the reader is referred to [2,3], where the authors investigate qualitative aspects of global solutions of hyperbolic Kirchhoff systems, both in the classical framework and in a more general setting given by anisotropic Lebesgue and Sobolev spaces.In particular it is shown that a global solution u converges to an equilibrium state in the sense of the energy decay, provided that the initial data are sufficiently small.Not all local solutions of (1.1) are global in time.For the weakly damped case(ω = 0, µ > 0, m = 1), Pucci and Serrin [21] proved nonexistence of global solutions when E(0) < d and u 0 ∈ N − .In the case when ω > 0 and µ = 0. Ono [19] showed that the solution of (1.1) blow up in finite time if E(0) < 0, which automatically implies u 0 ∈ N + .Ohta [18] improves this result by allowing E(0) < d and u 0 ∈ N + .Gazzola and Squassina [8] extended this result to the case when µ = 0 and E(0) ≤ d.All those works mentioned above dealt with the linear damping case (m = 1) or when the weak damping is absent(µ = 0).In the case of (1.1) with m > 1 however, the most frequently used technique in the proof of blow up named "concavity argument" no longer apply, so it is necessary to use another approach, namely the blow up theorem 2.3 in [17] for all negative initial energies.In the recent paper [4], thanks to a new combination of the potential well and concavity methods, the global nonexistence of solutions has been proved for Kirchhoff systems when ω = 0 and the initial energy is possibly above the critical level d.
The paper is organized as follows.In section 2, we state the local existence result and recall some notations and useful lemmas.In section 3, we present the boundedness result of global solutions under the assumptions E(t) ≥ d and m < p.In section 4, we state a sufficient and necessary condition on which the solution of (1.1) is global.In section 5, blow up behavior of (1.1) is investigated.In section 6, we present a exponential decay result.

Preliminaries
We specify some notations first.In this context, we denote • q by the L q norm for 1 ≤ q ≤ ∞, and ∇u 2 the Dirichlet norm of u in H 1 0 (Ω).We define the C 1 functionals I, J, E: H Note that E(t) satisfies the energy identity where T max is the maximal existence time of u(t).The mountain pass level of J is defined as Denote the best sobolev constant for the embedding We introduce the sets And the Nehari manifold N is defined by which intersects H 1 0 (Ω) into two unbounded sets We also consider the sublevels of J and we introduce the stable set S and the unstable set U defined by where S(t) denotes the corresponding semigroup on H 1 0 (Ω) × L 2 (Ω), generated by problem (1.1).Moreover, if We restricted ourselves to the case ω > 0, µ = 0 and N ≥ 3, the other cases being similar.For a given T > 0, we choose the work space H = C([0, T ]; 2 ).We divide the proof the local existence theorem into two lemmas.
Lemma 2.3 For every T > 0, every w ∈ H and every initial data (u 0 , u 1 ) satisfies (1.3), there (2.5) Proof.Existence.We consider a standard Galerkin approximation scheme for the solution of (2.5) based on the eigenfunction {e k } ∞ k=1 of the operator −∆ with null boundary condition on ∂Ω.That is, we let for all v ∈ V n := the linear span of {e 1 , e 2 , . . ., e n }, (•, •) denotes the standard L 2 (Ω) inner product.
By standard nonlinear ODE theory one obtains the existence of a global solution to (2.6) with the following a priori bounds uniformly in n We estimate the last term on the right-hand side Hence, there exists a subsequence of u n , which we still denoted by u n , such that The existence of u solving (2.5) is proved.
Uniqueness.If u 1 , u 2 are two solutions of (2.5) with the same initial data, set u = u 1 − u 2 , substracting the equations and test with u t , we obtain Observe that g(u) = |u| m−1 u is increasing, we immediately get u 1 = u 2 .The proof of the lemma is complete.
Denote F the mapping defined by the equation (2.5), i.e., u = F (w Proof.By Lemma 2.3, for any given w ∈ B R , the corresponding solution satisfies the following energy equality We estimate the last term on the right-hand side by using Hölder, Young's inequality and Sobolev embedding theorem where Combining (2.9) with (2.10), by choosing T sufficiently small, we get Observe that for any given ball K ⊆ H, any solution to (2.5) with w ∈ K with finite initial energy must satisfy The above inequality and Simon's compactness lemma imply the compactness of F (K).We need only to prove that For this purpose, take w 1 , w 2 ∈ B R , substracting the two equations (2.5) for u 1 = F (w 1 ) and and then we obtain for all η ∈ H 1 0 (Ω), take η = u t , integrate the above equality over (0, t], notice that the last term on the left-hand side of the equality is nonnegative, we obtain The proof of the lemma is complete. , where θ = 2N (p + 1 − q)/((p + 1)(2N + 2q − N q)).
Lemma 2.6 Under the assumptions of Lemma 2.5, the following inequalities hold The proof of the above two lemmas are elemental, so we omit it.
3 Boundedness of Global Solutions Lemma 3.1 Assume that E(t) ≥ d for all t ≥ 0, then for every t ≥ 0, there exists a positive constant C, such that Theorem 3.2 Assume that ω > 0, let m < p and E(t) ≥ d for all t ≥ 0, then every global solution to (1.1) is bounded.Moreover, if n = 1, 2 or if n ≥ 3 and 1 < p < n+2 n−2 , then there exists a positive constant l such that S l = ∅, Proof.According to [8], the difficult part is to prove the boundedness of global solution.Once the boundedness result is established, the convergence up to a sequence of solutions of (1.1) towards a steady-state result of the theorem can be arrived by following the same arguments as in [8]  Taking into account that u t (τ ) ∈ H 1 0 (Ω) for a.e.τ ≥ 0, combine Poincaré inequality with the energy equality (2.1), we have for every t > 0, Letting t → ∞, we can conclude It is easy to observe from the above inequality that, for every t ≥ 0, there exists a positive constant Furthermore, by the definition of E(t), we can obtain , inspired by [7], we shall prove where C > 0 is a constant.
For this purpose we introduce the function where M > 0 to be specified later.Hence and from the energy equality, by applying Hölder and Young's inequality, in view of the convex property of the norm u m+1 m+1 , we have For this chosen ε, take M = max{ 1 2ε , m m+1 ε − 1 m }, then all the above inequalities hold.Take η = min{δ, 1} > 0, we get from (3.6) that Ḣ(t) ≥ η Ẽ(t) − (p + 1)E(0). (3.7) Integrate the above inequality over (t, t + 1) and then estimate the integral on the left-hand side, from Hölder inequality and (3.3), combining (3.7) with Lemma 3.1, the above inequality yields (3.5).
Following the proof of Theorem 2.8 [7], we can prove there exists a positive constant κ, such that for any 0 ≤ s ≤ t ≤ s + 1.
For the weakly damped case(ω = 0), we have the following Theorem 3.3 Assume that ω = 0, let m < p and E(t) ≥ d ∀ t ≥ 0, suppose further that Then every global solution to (1.1) is bounded.Moreover, if n = 1, 2 or if N ≥ 3 and 1 < p < N N −2 , then there exists a positive constant l such that S l = ∅, Similar proof can be done following the arguments of Theorem 3.2 by utilizing Lemma 3.1.

Global Existence
Theorem 4.1 Assume that (1.2) and (1.3) being fulfilled, and let u be the unique local solution to (1.1).If m ≥ p, then problem (1.1) admits a unique solution u(t, x) such that for any T > 0, EJQTDE, 2009 No. 39, p. 9 Proof can be done by following the arguments in [9].
which implies that T max = ∞ by virtue of Theorem 2.2.
It follows again from the energy identity (2.1) By integrating over [t 0 , t] the trivial inequality we have Since J(u) ≤ CI(u)(see [12] Lemma 2.5), the above inequality yields Moreover, by testing the equation (1.1) with u, we have for all t ∈ [t 0 , ∞), which implies By integrating the above equality over [t 0 , t], we have In view of [12] Lemma 3.4, we get .
To estimate the integral ω(∇u(t), ∇u t (t)), we use Hölder and Young's inequality To estimate the term µ Ω |u t (t)| m−1 u t (t)u(t)dx, we use Hölder inequality and interpolation in- where θ = ( ).In the above estimates, we used the equality followed from Lemma 2.5(ii), i.e., Since 1 − (p + 1)/(m + 1) − θ + (p + 1)θ/2 = 0, by using Young's inequality, we get from (5.3) that It follows from ( In view of the inequality where the constant σ > 2 will be chosen later. We obtain from (5.5) the inequality Choose the constant σ so that which is possible since E(t 0 ) < d, and this guarantees σ > 2.
Then, using this choice and (2.16) we obtain EJQTDE, 2009 No. 39, p. 12 For this chosen σ, we choose ε 2 small enough so that Finally, the inequality (5.6), Lemma 2.5 and Lemma 2.6 yield thus, the norm u(t) 2 has at least linear growth for t ≥ t 0 .On the other hand, we estimate the norm u(t) 2 from above.For t ≥ t 0 , we have where in the above estimates we used the Hölder inequality with respect to t, the boundedness of the integral t t 0 u t (τ ) m+1 m+1 dτ .Obviously, the inequality (5.9) contradicts with the inequality (5.8).Sufficiency.Suppose T max < ∞, then it follows from the last assertion of Theorem 2.2 that lim t→Tmax u(t) m+1 = ∞. (5.10) Observe the energy equality (2.1), we obtain Remark 5.4 It can be observed from the proof that the condition m < p was given for necessity, and p < 2(m + 1)/N + 1 was given for sufficiency.

Exponential Decay
In what follows, we shall assume, without loss of generality, that ω = µ = 1.
Then there exist positive constants C and β such that the global solution to problem (1.1) satisfies Lemma 6.2Under the assumptions of Theorem 6.1, we have for all t ≥ 0, u(t) ∈ N + .
Proof .Since I(u 0 ) > 0, there exists a T > 0, such that I(u(t)) ≥ 0 for all t ∈ [0, T ), which tells The Sobolev embedding theorem entails ≤ α, we can repeat the procedure and extend T to 2T , by continuing the argument and the lemma is so proved.
Proof of Theorem 6.1.We modify the function defined in Section 3 as follows We shall prove, for ε sufficiently small, there exist two positive constants c 1 and c 2 such that Take δ = ε, then choose ε small enough , we see there exists a c 1 > 0, such that G(t) ≥ c 1 (t Then we choose δ small enough to guarantee 2(p + 1) p − 1 γ(1 − λ) − δC > 0.
Now let us turn to the global existence of solutions starting with suitable initial data.Assume that (1.2) and (1.3) being fulfilled, and let u be the unique local solution to (1.1) as in Theorem 2.2.Then there exists a t 0 ∈ [0, T max ), such that u(t 0 ) ∈ S and E(u(t 0 )) < d if and only if T max = ∞ and lim As a byproduct of our proof, it is clear that under the restrictions on m and p, T max < ∞ if and only if E(t) → −∞ as t → T max .In particular, the blow up has a full characterization in terms of negative energy blow up.