## Differential and Integral Equations

### On classification of blow-up patterns for a quasilinear heat equation

#### Abstract

We study the asymptotic blow-up behavior of nonnegative solutions to the quasilinear heat equation $$u_t = (u^2)_{xx} + u^2 \quad \text{for x \in \mathbf{R}, \,\, t > 0},$$ with nonnegative, bounded, continuous initial data. We give a complete classification of all possible types of blow-up behavior for compactly supported initial data. For data which look like a step function we construct self-similar blow-up patterns (logarithmic traveling wave solutions) for which the corresponding blow-up sets are empty.

#### Article information

Source
Differential Integral Equations, Volume 9, Number 4 (1996), 655-670.

Dates
First available in Project Euclid: 7 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367969880

Mathematical Reviews number (MathSciNet)
MR1401430

Zentralblatt MATH identifier
0851.35057

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions

#### Citation

Bebernes, J.; Galaktionov, Victor A. On classification of blow-up patterns for a quasilinear heat equation. Differential Integral Equations 9 (1996), no. 4, 655--670. https://projecteuclid.org/euclid.die/1367969880