[1]李刚,于勇.半线性抛物方程可变号解的全局存在和爆破[J].东南大学学报(自然科学版),2005,35(5):829-832.[doi:10.3969/j.issn.1001-0505.2005.05.037]
 Li Gang,Yu Yong.Global existence and blow-up of sign-changing solutions to semilinear parabolic equations[J].Journal of Southeast University (Natural Science Edition),2005,35(5):829-832.[doi:10.3969/j.issn.1001-0505.2005.05.037]
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半线性抛物方程可变号解的全局存在和爆破()
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《东南大学学报(自然科学版)》[ISSN:1001-0505/CN:32-1178/N]

卷:
35
期数:
2005年第5期
页码:
829-832
栏目:
数学、物理学、力学
出版日期:
2005-09-20

文章信息/Info

Title:
Global existence and blow-up of sign-changing solutions to semilinear parabolic equations
作者:
李刚 于勇
南京信息工程大学数学系, 南京 210044
Author(s):
Li Gang Yu Yong
Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, China
关键词:
临界指数 可变号解 爆破
Keywords:
critical exponent sign-changing solutions blowup
分类号:
O175.26
DOI:
10.3969/j.issn.1001-0505.2005.05.037
摘要:
研究了半线性抛物方程ut=uxx+(1+t)q|u|p-1u/|x|σ1的初边值问题,σ≥0,q>σ/2-1,边值为零,初值满足某种速降条件.首先作自相似变换,再考虑相应的特征值问题,定义能量函数E(s),利用一些已有的不等式,最后运用无穷维的动力系统理论证明了问题的可变号解存在临界指数pk,pk=1+(2+2q-σ)/(k+1),k为某一自然数,k是初值在定义域D上的变号次数,当1<p≤pk,问题的所有非零解在有限时刻爆破; p>pk,问题存在一个非零全局解.
Abstract:
The equation: ut=uxx+(1+t)q|u|p-1u/|x|σ1 is considered, where σ≥0,q>σ/2-1, bound value being equal to zero, initial value being nonincreasing. Firstly, a self-similar transformation is made, then its eigenvalue problem is considered, energy function E(s) is defined, some existing inequalities are used, and the theory of infinite dimensional dynamical systems is adopted. It is shown that for any nonnegative integer k, k is the number of initial value sing-changing in bounded domain D,pk=1+(2+2q-σ)/(k+1) is the critical exponent for the above problem, i.e, if 1<p≤pk, then any nontrivial solution blows up at finite time; if p>pk, then the problem has a global solution.

参考文献/References:

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备注/Memo

备注/Memo:
基金项目: 江苏省教育厅自然科学研究计划资助项目(02KJB170002).
作者简介: 李刚(1958—),男,硕士,副教授,ligang@nuist.edu.cn.
更新日期/Last Update: 2005-09-20