## Differential and Integral Equations

### Existence of global solutions for a semilinear parabolic Cauchy problem

Munemitsu Hirose

#### 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

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