Differential and Integral Equations

Existence of global solutions for a semilinear parabolic Cauchy problem

Munemitsu Hirose

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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.

Article information

Source
Differential Integral Equations, Volume 21, Number 7-8 (2008), 623-652.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356038615

Mathematical Reviews number (MathSciNet)
MR2479684

Zentralblatt MATH identifier
1224.35135

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35K15: Initial value problems for second-order parabolic equations

Citation

Hirose, Munemitsu. Existence of global solutions for a semilinear parabolic Cauchy problem. Differential Integral Equations 21 (2008), no. 7-8, 623--652. https://projecteuclid.org/euclid.die/1356038615


Export citation