Existence and nonexistence of global solutions to the Cauchy problem of the nonlinear hyperbolic equation with damping term

Abstract: This paper concerns with the Cauchy problem for two classes of nonlinear hyperbolic equations with double damping terms. Firstly, by virtue of the Fourier transform method, we prove that the Cauchy problem of a class of high order nonlinear hyperbolic equation admits a global smooth solution u(x, t) ∈ C∞((0,T ]; H∞(R)) ⋂C([0,T ]; H3(R)) ⋂C1([0,T ]; H−1(R)) as long as initial value u0 ∈ W4,1(R) ⋂ H3(R), u1 ∈ L1(R) ⋂ H−1(R). Moreover, we give the sufficient conditions on the blowup of the solution of a nonlinear damped hyperbolic equation with the initial value conditions in finite time and an example.


Introduction
In 1997, Banks et al. [1] established a class of nonlinear damped hyperbolic equation (1.1) As a model it describes the motion of a neo-Hooken elastomer rod with internal damping, where A 2 w t is the exact form of the internal dynamic damping mechanisms in elastomers, A 1 , A 2 , N and f satisfy certain assumptions. When A 1 = A 2 = − ∂ 2 ∂x 2 , N = − ∂ ∂x , Equation (1.1) becomes The model is well known and it is been described the dynamical longitudinal vibrations of an neo-Hooken material rod, and there have been many researches on the global existence and blow-up of solutions for Equation (1.3) (see [2,3]).
Equation (1.4) describes the propagation of the wave in the medium with the dispersion effect, and it is connected with the equations in [4]- [10]. In [11], Yang et al. proved the well-posedness of Cauchy problem for the nonlinear beam system (1.4). When g(s) = s n , n ≥ 2, they proved the global existence of smooth solutions as long as initial data ϕ ∈ L 2 (R) H 2 (R), ψ ∈ L 1 (R) L 2 (R). In [12], Chen et al. Equation (1.5) models the vibration of a nonlinear damped beam with fixed boundary, taking account of the internal material damping. Banks et al. [13] established the existence and uniqueness of the global weak solutions to the initial boundary value problem of Equation (1.5). Later, Ackleh et al. [14] studied such system (1.5) to find the existence of weak solutions of the mixed problem in a bounded domain. In [15], Chen et al. gived the sufficient conditions of blow-up result for a nonlinear damped hyperbolic equation. Further generalizations are also given in [16]- [19] and the references therein. When A 1 = ∂ 4 ∂x 4 , A 2 = − ∂ 2 ∂x 2 + ∂ 4 ∂x 4 , N = − ∂ 2 ∂x 2 , f = 0, Equation (1.1) becomes u tt − u xxt + u xxxx + u xxxxt = g(u xx ) xx , x ∈ Ω, t > 0. (1.6) Recently, Yu et al. [20] established the existence and nonexistence of the global weak solutions to the initial boundary value problem of a nonlinear beam equation with double damping terms (1.6) provded that Obviously, conditions (1.7) and (1.8) imply that the growth order of the nonlinear term g(s) is not more than 1. The reason for the strong assumptions (1.7) and (1.8) lie in that it is very difficult to dominate the effect of the nonlinear term g(u xx ) xx by standard a priori estimate technology.
where u(x, t) denotes the unknown function, k 1 is a positive constant. g(s) is a given nonlinear function, u xxxxt denotes the strong material damping and u xxt represents the internal dynamic damping. As far as we know, there is little research on analysis of Equation (1.9) with material damping and internal dynamic damping at the same time. In this case, what happens to the existence and nonexistence of global solution to the problem (1.9)-(1.10) remain open. When where a 2 u x 8 is a "good" regular term. Obviously, g(s) is monotone if and only if n is an odd number. For this kind of g, does problem (1.11), (1.10) have any global solutions for initial data belonging to suitably chosen functional spaces? On the other hand, if g(s) is not monotone and g (s) is not bounded below, say g(s) = s 2m (where m ≥ 1 is an integer), does the initial value problem (1.11), (1.10) have any global solutions? The question is interesting and open. This paper is organized as follows. In Section 2, the main results are stated. In Section 3, we prove the existence of global smooth solutions for the Cauchy problem (1.11), (1.10) by the Fourier transform method. In Section 4, using the modified concavity method, the sufficient conditions of blow-up of the solution for the Cauchy problem (1.9)-(1.10) will be given and we give an example to examine Theorem 2.3.
Suppose that the maximal time of existence of the solution of the Cauchy problem (1.9)-(1.10) is infinite. The energy functional for the problem (1.9) can be defined as Integration of (4.2) from 0 to t leads to We now define where β ≥ 0, T 0 and t 0 are positive real numbers to be given later. Hence,
such that This is a contradiction with the fact that the maximal time of the existence of the solution is infinite.
(2) If E(0) = 0, we choose β = 0 Then (4.11) becomes We take T 0 such that By considering the assumption (ii), we obtain φ(0) > 0, φ (0) > 0. Then according to Lemma 4.1 (1), , such that This is a contradiction with the fact that the maximal time of the existence of the solution is infinite.

Discussion
It is well known that Equation (1.11) describes the motion of the elastomer rod with internal damping. In the process of high speed movement, by the impact on damping characteristic and external excitation, the state of the elastomer rod is complicated and unpredictable at the initial velocity. Considering this situation, we choose initial data belonging to more general functional space u 0 ∈ W 4,1 (R) H 3 (R), u 1 ∈ L 1 (R) H −1 (R). By using the L 1 −based spaces instead of L 2 −based ones, which are completely different from those used in [11], [18], we can still obtain the global smooth solution in the generalized space. In this paper, we just consider the problems in 1-dimensional space, but in high-dimensional space, do both Equation (1.9) and Equation (1.11) have and global solutions to the Cauchy problem or the initial boundary value problem? The question is interesting and opening.