Open Access
2008 Existence of global solutions for a semilinear parabolic Cauchy problem
Munemitsu Hirose
Differential Integral Equations 21(7-8): 623-652 (2008). DOI: 10.57262/die/1356038615


In this paper, we consider the parabolic equation $w_t=\Delta w+|x|^{l} w^p$, $x \in {\bf R}^n$, $t>0$ with $w(x, 0)=f(x)$ and show the existence of global solution if $1+(2+l)/n < p <(n+2+2l)/(n-2)$ for each $n \geq 3$ and $l \in (-2, l^*]$, where $l^*=0$ if $n \geq 4$ and $l^*=\sqrt3-1$ if $n=3$. In order to prove this result, we need an upper solution for this Cauchy problem. If $f(x)$ satisfies some condition, then we can show the existence of upper solution by investigating the structure of positive radial solutions for related elliptic equation which has a gradient term.


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Munemitsu Hirose. "Existence of global solutions for a semilinear parabolic Cauchy problem." Differential Integral Equations 21 (7-8) 623 - 652, 2008.


Published: 2008
First available in Project Euclid: 20 December 2012

zbMATH: 1224.35135
MathSciNet: MR2479684
Digital Object Identifier: 10.57262/die/1356038615

Primary: 35K55
Secondary: 35K15

Rights: Copyright © 2008 Khayyam Publishing, Inc.

Vol.21 • No. 7-8 • 2008
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