## Differential and Integral Equations

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

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

J. Bebernes and Victor A. Galaktionov

#### 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