## Differential and Integral Equations

- Differential Integral Equations
- Volume 9, Number 4 (1996), 671-700.

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

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367969881

**Mathematical Reviews number (MathSciNet)**

MR1401431

**Zentralblatt MATH identifier**

0853.35067

**Subjects**

Primary: 35L50: Initial-boundary value problems for first-order hyperbolic systems

#### Citation

Secchi, Paolo. The initial-boundary value problem for linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity. Differential Integral Equations 9 (1996), no. 4, 671--700. https://projecteuclid.org/euclid.die/1367969881