河北师范大学学报—

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刊名：河北师范大学学报（自然科学版）Journal of Hebei Normal University (Natural Science)
主办：河北师范大学
ISSN： 1000-5854
CN： 13-1061/N
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具有位势项的热量方程解的全局存在性与爆破现象

广东财经大学华商学院数据科学学院, 广东广州 511300
DOI： 10.13763/j.cnki.jhebnu.nse.202101010

The Existence and the Blow-up Phenomenon of Solutions to the Heat Equation with Variable Coeffcients Under Nonlinear Boundary Conditions

LI Zhiqing, LI Yuanfei, LI Zongkai

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摘要/Abstract

摘要：

考虑了定义在Ω⊂R^N（N ≥ 2）上的热量方程u_t=Δu^m-V（x）u+u^p，m，p>1，当m+1>2p时，证明了解的全局存在性.若m<p，假设方程的初始数据满足一定的约束条件，证明了方程的解在有限时刻一定发生爆破并获得了爆破时间的上界.如果2 < N < 4p/2p-（m+1）且p < m+1 < 2p，或者N=2且p+1 < m+1 < 2p，确定了爆破时间的下界.

Abstract：

In this paper, the heat equation with variable coefficients defined on Ω is considered, in which Ω⊂R^N(N ≥ 2) is a bounded convex region and the equation has nonlinear boundary conditions. By using differential inequalities, we first derive the conditions under which the blow-up must occur and determine the upper bound of the blow-up time. Meanwhile,by making certain restrictions on the nonlinear terms,we also obtain the global existence of the solution. When the blow-up occurs,the lower bound of blasting time is also determined.

关键词

关键词： 热量方程 ; 全局解 ; 爆破 ; 上界

Key words： heat equation ; global solution ; blow-up ; upper bound

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