## Differential and Integral Equations

### On bounds for global solutions of semilinear parabolic equations with critical and subcritical Sobolev exponent

Michinori Ishiwata

#### Abstract

In this paper, we are concerned with the existence of $L^\infty$-global bounds for global-in-time solutions of some semilinear parabolic problems. It is well known that every global-in-time solution for the subcritical problem is globally bounded in $L^\infty$, while there exists a global solution which is not bounded in $L^\infty$ globally in time in the critical case. In this paper, we discuss the necessary and sufficient condition for the existence of $L^\infty$-global bounds, which is valid for the subcritical and the critical case in a unified way. Moreover, using our main results, we provide various examples with the critical exponent in which every global-in-time solution has an $L^\infty$-global bound.

#### Article information

Source
Differential Integral Equations, Volume 20, Number 9 (2007), 1021-1034.

Dates
First available in Project Euclid: 20 December 2012

https://projecteuclid.org/euclid.die/1356039309

Mathematical Reviews number (MathSciNet)
MR2349378

Zentralblatt MATH identifier
1212.35212

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B33: Critical exponents 35B45: A priori estimates

#### Citation

Ishiwata, Michinori. On bounds for global solutions of semilinear parabolic equations with critical and subcritical Sobolev exponent. Differential Integral Equations 20 (2007), no. 9, 1021--1034. https://projecteuclid.org/euclid.die/1356039309