一类波动方程整体解的存在性与不存在性

doi:10.3969/j.issn.1000-5641.2018.02.001

华东师范大学学报(自然科学版) ›› 2018, Vol. 2018 ›› Issue (2): 1-10.doi: 10.3969/j.issn.1000-5641.2018.02.001

• 数学 • 下一篇

一类波动方程整体解的存在性与不存在性

晋守博, 张祖峰

宿州学院数学与统计学院, 安徽宿州 234000

收稿日期:2017-03-14 出版日期:2018-03-25 发布日期:2018-03-22
作者简介:晋守博,男,副教授,研究方向为偏微分方程.E-mail:jin_shoubo@163.com.
基金资助:
安徽省自然科学研究项目（1508085MA10）；安徽省高校自然科学研究重点项目（KJ2016A770）；宿州学院重点科研项目（2016yzd06）；宿州学院优秀青年人才支持计划重点项目（2016XQNRL003）

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

JIN Shou-bo, ZHANG Zu-feng

School of Mathematics and Statistics, Suzhou University, Suzhou Anhui 234000, China

Received:2017-03-14 Online:2018-03-25 Published:2018-03-22

摘要/Abstract

摘要： 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.

关键词: 波动方程, k阶拉普拉斯算子, 存在性, 不存在性, 位势井

Abstract: In this paper we investigated the initial boundary value problem for a class of 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.

Key words: wave equation, k-Laplace operator, existence, non-existence, potential well

中图分类号:

O175.29

晋守博, 张祖峰. 一类波动方程整体解的存在性与不存在性[J]. 华东师范大学学报(自然科学版), 2018, 2018(2): 1-10.

JIN Shou-bo, ZHANG Zu-feng. Existence and non-existence of global solutions for the wave equations[J]. Journal of East China Normal University(Natural Sc, 2018, 2018(2): 1-10.

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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

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