基于交叉变分的非线性Klein-Gordon方程解的整体存在和爆破

徐润章, 张明有, 姜晓丽, 王雪梅, 沈继红

数学学报 ›› 2014, Vol. 57 ›› Issue (3) : 427-444.

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数学学报 ›› 2014, Vol. 57 ›› Issue (3) : 427-444. DOI: 10.12386/A2014sxxb0042
论文

基于交叉变分的非线性Klein-Gordon方程解的整体存在和爆破

    徐润章1, 张明有2, 姜晓丽1, 王雪梅1, 沈继红1
作者信息 +

Global Existence and Blow up of Nonlinear Klein-Gordon Equation Based on Cross Variational Method

    Run Zhang XU1, Ming You ZHANG2, Xiao Li JIANG1, Xue Mei WANG1, Ji Hong SHEN1
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文章历史 +

摘要

研究非线性Klein-Gordon方程的初边值问题,运用位势井方法,在E(0)<d的情况得到了方程解的整体存在和爆破.在临界能量状态得到了整体解的存在性与不存在性.最后使用凸性方法,得到某些具有高初始能量解的爆破.

Abstract

This paper discusses the initial boundary value problem of the nonlinear Klein-Gordon equation. By the potential well method, we obtain the global existence and blow up of solutions when the initial energy is less than the mountain pass level value. Moreover, the global existence and nonexistence of solutions are derived under the critical initial energy case. In the end, by using the concavity method, we establish a blow up result for certain solutions with arbitrary positive initial energy.

关键词

非线性Klein-Gordon方程 / 最佳条件 / 不变集合 / 整体存在 / 爆破

Key words

nonlinear Klein-Gordon equation / sharp condition / invariant manifold / global existence / blow-up

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导出引用
徐润章, 张明有, 姜晓丽, 王雪梅, 沈继红. 基于交叉变分的非线性Klein-Gordon方程解的整体存在和爆破. 数学学报, 2014, 57(3): 427-444 https://doi.org/10.12386/A2014sxxb0042
Run Zhang XU, Ming You ZHANG, Xiao Li JIANG, Xue Mei WANG, Ji Hong SHEN. Global Existence and Blow up of Nonlinear Klein-Gordon Equation Based on Cross Variational Method. Acta Mathematica Sinica, Chinese Series, 2014, 57(3): 427-444 https://doi.org/10.12386/A2014sxxb0042

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基金

国家自然科学基金(11101102);中国博士后基金(2013M540270);中央高校基本科研业务费专项资金;黑龙江省博士后基金及黑龙江省普通高等学校青年学术骨干支持计划(1252G020)
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