一类波动方程整体解的存在性与不存在性

Existence and non-existence of global solutions for the wave equations

研究了一类具有异号项的高阶波动方程初边值问题.首先介绍了这类方程的最新研究进展,并引入了几个重要的广义泛函和集合,然后讨论了这些泛函的性质,并证明了这些集合在该波动方程下是不变的.最后利用Galerkin逼近法和位势井法相结合证明了方程整体弱解的存在性,并利用位势井-凸性方法分析了方程整体弱解不存在的前提条件.同时给出了方程整体弱解存在与不存在的最佳门槛结果.

In this paper we investigated the initial boundary value problem for a classof higher order wave equations with two opposite source terms. Firstly, we introduced the latest research progress of the wave equations and defined some important generalized functionals and sets, then the properties of the functionals were discussed. Secondly, it was proved that these sets were invariant under the wave equation. Finally, we proved the existence of global weak solutions by the combination of Galerkin approximation method and potential well method, and obtained the conditions of the non-existence of global weak solutions by using the potential well method and the convexity. The optimal threshold results were given for the existence and non-existence of global weak solutions.