GLOBAL EXISTENCE AND ENERGY DECAY ESTIMATE OF SOLUTIONS FOR A CLASS OF NONLINEAR HIGHER-ORDER WAVE EQUATION WITH GENERAL NONLINEAR DISSIPATION AND SOURCE TERM

. This paper deals with a higher-order wave equation with general nonlinear dissipation and source term u (cid:48)(cid:48) + ( − ∆) m u + g ( u (cid:48) ) = b | u | p − 2 u, which was studied extensively when m = 1 , 2 and the nonlinear dissipative term g ( u (cid:48) ) is a polynomial, i.e., g ( u (cid:48) ) = a | u (cid:48) | q − 2 u (cid:48) . We obtain the global existence of solutions and show the energy decay estimate when m ≥ 1 is a positive integer and the nonlinear dissipative term g does not necessarily have a polynomial grow near the origin.


JUN ZHOU
For m = 1 and g(s) = as (a > 0), problem (1) was firstly studied by Levine [6,7]. He showed that solutions with negative initial energy blow up in finite time. Ikehata and Suzuki [4] showed that for sufficiently small initial data (u 0 , u 1 ), the solution trajectory (u(t), u (t)) tends to (0, 0) in H 1 0 (Ω) × L 2 (Ω) as t → ∞. When m = 1 and g(s) = a|s| r−2 s (r ≥ 2), Georgiev and Todorova [3] showed that if the damping term dominates over the source, then a global solution exists for any initial data. By using the stable set method due to Sattinger [13], Ikehata [5] proved that a global solution exists with no relation between p and r, and Todorova [14] proved that an energy decay rate is E(t) ≤ (1 + t) −2/(r−2) for t ≥ 0, for which she used the general method on energy decay introduced by Nakao [10]. Aassila [1] proved the existence of a global decaying H 2 (Ω) solution when g(s) has not necessarily a polynomial growth near zero, but with small parameter b.
When m = 2 and g(s) = a|s| r−2 s (r ≥ 2), Messaoudi [9] showed the solution of problem (1) blows up in finite time if p > r in the case that the initial energy is positive. On the other hand, he proved the solution is global in time if r ≥ p. Wu and Tsai [15] showed that the solution if global under some conditions without the relation between p and r. Moreover, they showed that the solution decays exponentially if r = 2 whereas the decay is of a polynomial order if r > 2. They also proved that the local solution blows up in finite time if p > r and the initial energy is nonnegative.
For general m ≥ 2 and g(s) = a|s| r−2 s (r ≥ 2), problem (1) was studied in [16,18]. Ye [16] showed the solution exists global if the initial energy is sufficiently small. Zhou et al. [18] proved the global existence result without the relation between p and r and showed that the energy functional decays algebraically by the method introduced by Nakao [10]. Moreover, the blow-up properties of the local solution with non-positive initial energy as well as small positive initial energy were also established. Unfortunately this method does not seem to be applicable to the case of more general functions g.
Our purpose in this paper is to give a global solvability in the class H m 0 (Ω) ∩ H 2m (Ω) and energy decay estimates of the solutions for problem (1) for a general nonlinear damping g. We use some new techniques introduce in [8] to derive a decay rate of the solutions. So we use the argument combining the method in [8] with the concept of stable set in H m 0 (Ω). We conclude this section by stating our plan and giving some notations. In Section 2 we formulate some lemmas need for our arguments. Sections 3 and 4 are devoted the proof of global existence and decay estimates for the problem (1). Throughout this paper all the functions considered are real-valued. We omit the space variable x of u(t, x), u t (t, x) and simply denote u(t, x), u t (t, x) by u(t), u (t), respectively, when no confusion arises. For 1 ≤ γ < ∞, · γ denotes the L γ (Ω) norm and we write · 2 by · for simplicity.

2.
Preliminaries. Firstly, we state a local existence result of problem (1), which can be obtained in a similar way as done in [2,9,11].
. Moreover, at least one of the following statements holds true: Let u(t) be the solution of (1) got in Theorem 2.1, we introduce some functional as in [18] Next we state three well known lemmas that will be needed later.
Then there is a constant C λ depending on Ω and λ such that Assume that there exist σ ≥ 0 and ω > 0 such that Then there exists C * > 0 depending on E(1) such that Lemma 2.4. [8, (6.23)-(6.25)] Assume (H2) holds. Then there exists a strictly increasing function φ : R + → R + satisfying the following conditions 3. Global existence. Since g is an odd increasing function, a direct calculation shows that Hence the energy is non-increasing and in particular E(t) ≤ E(0) for all t ≥ 0. Let The main result of this section is the following theorem: Theorem 3.1. Assume the assumptions in Theorem 2.1 hold. Let u(t) be the solution of problem (1) with initial data satisfying If E(0) satisfies where C p is the positive constant defined in Lemma 2.2, then u(t) ∈ H for all t ∈ [0, ∞), and there exists a positive constant M depending on E(0) such that which implies u(t) exists globally.
The above discussions imply u(t) ∈ H for t ∈ [0, T ). Next, we prove T = ∞ and (6) to complete the proof. By the fact that the energy E(t) is non-increasing, we have for t ∈ [0, T ) since I(t) ≥ 0, and hence The above estimate implies T = ∞. Furthermore, it follows from (2) that where G(t) := tg(t), C * 1 is a positive constant only depending on initial energy E(0) in a continuous way, s 1 ≥ max{1, 1 G −1 (1) } be such that G(1/s 2 ) ≤ 1. If, in addition, t → g(t)/t is non-decreasing in [0, 1] and g (0) = 0, then we have where C * 2 is a positive constant only depending on initial energy E(0) in a continuous way, s 2 ≥ max{1, 1 g −1 (1) } be such that g(1/s 2 ) ≤ 1.
Then it follows from (26) that .