非线性扩散方程Cauchy问题的临界指数

数学物理学报 ›› 2013, Vol. 33 ›› Issue (5): 966-976.

非线性扩散方程Cauchy问题的临界指数

李中平|杜宛娟|穆春来

西华师范大学数学与信息学院四川南充 637002；重庆大学数学与统计学院重庆 400044

收稿日期:2012-03-08 修回日期:2013-06-23 出版日期:2013-10-25 发布日期:2013-10-25
基金资助:
国家自然科学基金面上项目(11071266)、国家自然科学基金数学天元项目(11226181)和西华师范大学科研项目(12B024, 12A032)资助.

Critical Exponents of the Cauchy Problem for Nonlinear Diffusive Equations

LI Zhong-Ping, DU Wan-Juan, MU Chun-Lai

College of Mathematic and Information, China West Normal University, Sichuan Nanchong 637002； College of Mathematics and Statistics, Chongqing University, Chongqing 400044

Received:2012-03-08 Revised:2013-06-23 Online:2013-10-25 Published:2013-10-25
Supported by:
国家自然科学基金面上项目(11071266)、国家自然科学基金数学天元项目(11226181)和西华师范大学科研项目(12B024, 12A032)资助.

摘要/Abstract

摘要：

主要研究非线性扩散方程u_t=div(|nabla u|^p-2 nabla u)+|x|^σu^q在非平凡、非负初始条件下的大时间行为. 这里p>2, σ>0及q>p-1. 证明了Fujita临界指数q_c=p-1+p+σ /N. 即证明了: 如果q<q_c, 则所有正解都在有限时刻爆破, 但是当q>q_c时, 全局解和非全局解都有可能存在. 而且还根据初值在无穷远处的衰减行为建立了第二临界指数.

关键词: Fujita临界指数, 第二临界指数, 非线性扩散方程

Abstract:

In this paper, we investigate the large time behavior of solution to nonlinear diffusive equations u_t=div(|nabla u|^p-2 nabla u)+|x|^σu^q with nontrivial, nonnegative initial data. Here p>2, σ>0 and q>p-1. We prove that q_c=p-1+p+σ /N is the critical Fujita exponent. That is, if q<q_c then every positive solution blows up in finite time, but for q>q_c, there exist both global and non-global solutions to the problem. Furthermore, we establish the secondary critical exponent on the decay asymptotic behavior of an initial value at infinity.

Key words: Critical Fujita exponent, Secondary critical exponent, Nonlinear diffusive equations

中图分类号:

35K50

李中平, 杜宛娟, 穆春来. 非线性扩散方程Cauchy问题的临界指数[J]. 数学物理学报, 2013, 33(5): 966-976.

LI Zhong-Ping, DU Wan-Juan, MU Chun-Lai. Critical Exponents of the Cauchy Problem for Nonlinear Diffusive Equations[J]. Acta mathematica scientia,Series A, 2013, 33(5): 966-976.

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