一类具耗散项的非线性四阶波动方程的整体弱解及其渐近性质

Global Existence and Decay Estimate to the Initial-Boundary Value Problem of the Nonlinear Fourth-order Dissipative Wave Equation

本文讨论一类具耗散项的非线性四阶波动方程的初边值问题.该问题来源于不同的物理背景,例如平面中固定金属板的运动、梁的振动,以及水波的相互作用等都涉及这一问题.利用位势井理论和紧致性方法,我们证明了当初始能量为正但有适当上界,非线性项满足假设条件时,该问题整体弱解的存在性.并在此基础上,利用方程中耗散项的作用和一个微分不等式得到了解的渐近性质.

The initial-boundary value problem of the nonlinear four-order wave dissipative equation is investigated in this paper. The problem is derived from diverse physical background such as the study of plate and beams and the study of interaction of water waves. We prove the existence of the global solution to this problem by means of the compactness method and the potential well idea. Meanwhile, we obtain the decay estimate of the energy of the global solution to this problem by employing the role of an important dissipative term and a difference inequality.