## Differential and Integral Equations

### The initial-boundary value problem for linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity

Paolo Secchi

#### 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), 671-700.

Dates
First available in Project Euclid: 7 May 2013