In this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

中文翻译：

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一类高阶n维非线性伪抛物方程的初边值问题

在本文中，我们研究了一类不具有正能量并且来自土壤力学，热传导和非线性光学的高阶n维非线性伪抛物方程的初边值问题。通过山口定理，我们首先证明了静态问题的非零弱解的存在，这是演化问题的重要基础，然后基于势阱的方法，我们证明了演化问题的整体弱解的存在。