一类带有非线性阻尼项和源项的四阶波动方程整体解的存在性与不存在性

数学物理学报 ›› 2015, Vol. 35 ›› Issue (3): 618-633.

一类带有非线性阻尼项和源项的四阶波动方程整体解的存在性与不存在性

狄华斐¹, 尚亚东^1,2

1. 广州大学数学与信息科学学院广州 510006;
2. 广东省教育厅数学与交叉科学重点实验室广州 510006

收稿日期:2014-01-20 修回日期:2014-11-23 出版日期:2015-06-25 发布日期:2015-06-25
作者简介:狄华斐, dihuafei@yeah.net;尚亚东, gzydshang@126.com
基金资助:
国家自然科学基金(11401122, 40890150)和广东省科技计划项目(2008B080701042)资助

Global Existence and Nonexistence of Solutions for A Class of Fourth Order Wave Equation with Nonlinear Damping and Source Terms

Di Huafei¹, Shang Yadong^1,2

1. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006;
2. Key Laboratory of Mathematics and Interdisciplinary Sciences of GuangdongHigher Education Institutes, Guangzhou University, Guangzhou 510006

Received:2014-01-20 Revised:2014-11-23 Online:2015-06-25 Published:2015-06-25

摘要/Abstract

摘要：

考虑一类带有非线性阻尼项和源项的四阶波动方程的初边值问题. 通过结合 Galerkin 逼近, 势井方法和单调紧致方法, 在最少的先验估计下获得了整体解的存在性. 此外, 在初始能量为负的情况下, 证明了存在有限时间内爆破的解.

关键词: 波动方程, 整体存在性, 爆破, 阻尼项和源项, 单调紧致法

Abstract:

In this paper, we consider the initial boundary value problem of the fourth order wave equation with nonlinear damping and source terms. By the combination of Galerkin approximations, Potential well and Monotonicity-Compactness methods, the global existence of solutions is obtained with the least amount of a priori estimates. Moreover, we prove that there are solutions with negative initial energy that blow up in finite time.

Key words: Wave equation, Global existence, Blow up, Damping and source terms, Monotonicity-Compactness methods

中图分类号:

O175.4

狄华斐, 尚亚东. 一类带有非线性阻尼项和源项的四阶波动方程整体解的存在性与不存在性[J]. 数学物理学报, 2015, 35(3): 618-633.

Di Huafei, Shang Yadong. Global Existence and Nonexistence of Solutions for A Class of Fourth Order Wave Equation with Nonlinear Damping and Source Terms[J]. Acta mathematica scientia,Series A, 2015, 35(3): 618-633.

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