THE INITIAL-BOUNDARY VALUE PROBLEMS FOR A CLASS OF SIXTH ORDER NONLINEAR WAVE EQUATION

. This paper considers the initial boundary value problem of solutions for a class of sixth order 1-D nonlinear wave equations. We discuss the probabilities of the existence and nonexistence of global solutions and give some suﬃcient conditions for the global and non-global existence of solutions at three diﬀerent initial energy levels, i.e., sub-critical level, critical level and sup-critical level.

1. Introduction. In this paper, we consider the initial boundary value problem (IBVP) for the following 1-D nonlinear wave equation of sixth order u(0, t) = u(1, t) = u xx (0, t) = u xx (1, t) = 0, t ∈ (0, ∞), where f (u) = ±|u| p or |u| p−1 u, p > 1 is a constant, u 0 and u 1 are given initial data, a > 0 is a given constant satisfying certain conditions to be specified later.
In the study of a weakly nonlinear analysis of elastoplastic-microstructure models for longitudinal motion of an elasto-plastic bar in [1] there arose the model equation x ) x , where u(x, t) is the longitudinal displacement, and α > 0, β = 0 are any real numbers. The author in [1] proved the instability of the special solution and the instability of the ordinary strain solution.
In the study of model for the propagation of small amplitude long waves on the surface of shallow water, there arose the classical Boussinesq type equation The equation (4) was first introduced by Boussinesq [2]. Later, extensive research has been carried out to study the Boussinesq equation by different views. Cho and Ozawa [3] established the global existence and the scattering of a small amplitude solution to the Cauchy problem of the equation (4). Considering the effect of damping, Varlamov [15,16] considered the following damped Boussinesq equation The local existence and the long-time decay of the initial value and initial boundary value problems with small initial data for one-dimensional space was studied in [15]. For two dimensional space, the long-time asymptotics of global solution for the initial boundary value problem in a ball can be seen in [16] Subsequently when Rosenau [13] was concerned with the problem of how to describe the dynamics of a dense lattice, he discovered equation (1) by a continuum method. Meanwhile one-dimensional homogeneous lattice wave propagation phenomena can also be described by equation (1). Wang and Xu in [19] considered the following Rosenau equation and proved global existence and blow up of the solution for the initial value problem with E(0) ≤ d (initial energy below the mountain pass level), where E(0) is the initial energy and d is the depth of the potential well defined. As for equation (5), Wang and Wang in [17] proved the decay and scattering of small-amplitude solution for γ = 1. Liu and Xu in [9] obtained the global existence and nonexistence of solution for the initial value problem of the equation (5) with E(0) ≤ d in the absence of u xxxxtt term. Later on Han and Chen [5] considered the initial boundary value problems for a class of nonlinear wave equations of the following form where a 1 , a 2 , a 3 > 0 are constants and φ(s) is a given nonlinear function. By the extension theorem they proved the existence and uniqueness of classical global solutions. Moreover sufficient conditions for blow up of solutions were also obtained.
Recently, for the Cauchy problem for equation (6) with f (s) = 0, Wang Yuzhu and Wang Yinxia [18] discussed the existence and nonexistence of global solutions to this problem under the case E(0) ≤ d. Shen et al. in [14] showed that the local solutions blow up in finite time under the high initial energy level E(0) > 0.
As the depth of potential well is very small even near zero, the range 0 < E(0) ≤ d is actually a very small range. Hence the properties of the initial data ensuring E(0) > 0 are of great interests. The task of the present paper is to consider almost all the possibilities of the initial data, i.e., 0 < E(0) < d (sub-critical case),E(0) = d (critical case) and E(0) > 0 (sup-critical case). For the sub-critical case and the critical case, we shall treat the problem for more general nonlinear terms such as f (u) = ±|u| p or |u| p−1 u.
In this paper, for the sub-critical initial energy case E(0) < d (Section 3), we first state a global existence of solution in the framework of the potential well method [12,8,21,20], and then inspired by the so-called concavity method [6,7] we prove a blow up result. For the critical initial energy case (E(0) = d) (Section 4), by utilizing the method of [10,20], we prove the global existence, finite time blow up of solutions. For the sup-critical initial energy case (E(0) > 0) (Section 5), by the ideas in [4,23,22] we derive some sufficient conditions on the initial data such that certain solutions exist globally or blow up in a finite time.
2. Some assumptions and preliminary lemmas. In this section we give some assumptions and preliminary results stating the main results of this paper. In what follows, we use the following abbreviations for simplicity of notation: First, we give some preliminary lemmas, then by using them we introduce the potential well W and the corresponding set V .
For problem (1)-(3) we introduce the potential energy functional the Nehari functional (8) and the energy functional Then from (7)- (9) we have Definition 2.2. We define , Proof. This lemma follows from which gives Parts (iv) and Part (v). Part (vi) follows from part (ii) and (11).

Corollary 1. If in Lemma 2.3 and Lemma 2.4 the assumption
, then the conclusions of Lemma 2.3 and Lemma 2.4 also hold.
Lemma 2.6. Let f (u) = ±|u| p or |u| p−1 u, u ∈ H and I(u) < 0, then a u x 2 + u xx 2 > r, where Proof. Note that I(u) < 0 implies a u x 2 + u xx 2 = 0. From this and we obtain a u x 2 + u xx 2 > r.
which together with a u x 2 + u xx 2 = 0 gives a u x 2 + u xx 2 ≥ r.
All nontrivial stationary solutions belong to the so-called Nehari manifold (see [11] and also [4]) defined by Then (12) (i) For f (u) = |u| p , we define Proof. Note that I(u) < 0 gives Hence from Lemma 2.4 it follows that J(λu) takes the maximum at λ * , which satisfies Let Then and a(u) < b(u).
Note (15) gives Therefore from Definition 2.8 we obtain Definition 2.10. We can define the stable set (potential well) and the unstable set In order to prove the main theorems, we first establish the local existence and uniqueness for solutions of problem (1)-(3).
By weak solution of problem (1) over [0, T max ], where T max is the maximum existence time of u, we mean a function u ∈ C([0, for all v ∈ H and a.e. t ∈ [0, T max ). Now we show a conservation law. holds.
Next, we present the following local existence theorem that can be established by combining arguments of [5] and [18].
3. Existence and nonexistence of global solutions at sub-critical energy level E(0) < d. In this section we discuss the invariance of the stable set (16) and the unstable set (17) Proof. We only prove the invariance of W , the proof for the invariance of V is similar. It follows from (10) and E(0) < d that J(u) < d. Then, if condition I(u 0 ) > 0 holds, we have u ∈ W for all t ∈ [0, T max ). Indeed, if it was false, there would exist a first time t 0 ∈ (0, T max ) such that I(u(t 0 )) = 0.
By the relation of C * and d, we get This contradicts J(u) < d. Thus, we have u ∈ W for all 0 ≤ t < T max . So the proof is completed.  Proof. From Theorem 2.12 and Lemma 2.11 it follows that problem (1)-(3) admits a unique local weak solution u ∈ C([0, T max ); H) satisfying where T max is the maximal existence time of u. From (8) and (18) we have From Lemma 3.1 we have u ∈ W for 0 ≤ t < T max , which says Again by Theorem 2.12 we obtain T max = +∞.
As M (t) is continuous on [0, T ], there exist constants δ 1 , δ 2 > 0 such that furthermore, and Testing the equation in (1) with u and plugging the result into the expression of M (t) we obtain From the definition of I(u), (24) becomes Therefore, we get where η : [0, T ] → R + is the function defined by Using Schwarz, Hölder and Young inequalities, we obtain These three inequalities entail η(t) ≥ 0 for every t ∈ [0, T ]. Therefore, we obtain for a.e. t ∈ [0, T ], where ξ : [0, T ] → R + is the map defined by By (10) and (14), (28) becomes Hence, there exists ρ > 0 such that By (21), (27) and (29) it follows that thus,   Proof. It follows from Theorem 2.12 that the problem (1)-(3) admits a unique local solution u ∈ C 1 ([0, T max ); H). In what follows, we prove that T max = ∞.
We prove this theorem by considering the following two cases.
The following lemma shows the invariance of the set V under the flow of problem (1)-(3).

5.
Existence and nonexistence of global solutions at high initial energy level E(0) > 0. In this section, we consider the existence and nonexistence of the global solutions for the problems (1)-(3) at high initial energy level E(0) > 0.
Suppose that E(0) > 0, the initial data satisfy and then the existence time of global solution for problem (1)- (3) is infinite.
Hence from Theorem 2.12 it follows that T max = ∞ and the solution of problem (1)-(3) exists globally.
In what follows, we show a preliminary lemma about the monotonicity of the functional u(x, t) 2 + u xx (x, t) 2 , which will be used to prove the invariance of the unstable set V under the flow of problem (1)-(3) at high initial energy level E(0) > 0.
Lemma 5.2. Let f (u) = ±|u| p or |u| p−1 u, u 0 (x), u 1 (x) ∈ H be given. Assume the initial data satisfy Let u(x, t) be the solution of equation (1) with initial data (u 0 , u 1 ). Then the map t → u(t) 2 + u xx (t) 2 is strictly increasing as long as u(x, t) ∈ V .