动态边界条件下一类强阻尼波动方程解的爆破

A Study of the Blow-Up for a Class of Strong-Damped Wave Equation with Dynamic Boundary Conditions

考虑了一类具有Kelvin-Voigt阻尼的波动方程.运用位势井理论,通过构造稳定集和不稳定集,结合能量分析的方法,首先证明了当初值属于稳定集时,该问题存在整体解;其次证明了强阻尼项的存在使得问题的解一致趋近于零,且具有指数衰减速率;最后给出了解在有限时间爆破的充分必要条件.

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions related to the Kelvin-Voigt damping.By combining the potential well method and energy analysis method and constructing stable and unstable sets,we prove that the global solution exists if the initial data are in the ＂stable set＂ and that under the presence of the strong damping the solutions uniformly approach zero at an exponential decay rate.Finally,we give the necessary and sufficient conditions for the solution blow up in finite time.