#### Campus Units

Mathematics

#### Document Type

Article

#### Publication Version

Published Version

#### Publication Date

1981

#### Journal or Book Title

SIAM Journal on Mathematical Analysis

#### Volume

12

#### Issue

6

#### First Page

893

#### Last Page

903

#### DOI

10.1137/0512075

#### Abstract

We consider the problem $u_{tt} = u_{xx} + \phi (u(x,t)),0 < x < L,t > 0;u(0,t) = u(L,t) = 0;u(x,0) = u_t (x,0) = 0$. Assume that $\phi :( - \infty ,A) \to (0,\infty )$ is continuously differentiable, monotone increasing, convex, and satisfies $\lim _{u \to A^ - } \phi (u) = + \infty $. We prove that there exist numbers $L_1 $ and $L_2 $, $0 < L_1 \leqq L$ such that if $L > L_2 $, then a weak solution *u* (to be defined) quenches in the sense that *u* reaches *A* in finite time; if $L < L_1 $, then *u* does not quench. We also investigate the behavior of the weak solution for small *L* and establish the local (in time) existence of *u*.

#### Copyright Owner

Society for Industrial and Applied Mathematics

#### Copyright Date

1981

#### Language

en

#### File Format

application/pdf

#### Recommended Citation

Chang, Peter H. and Levine, Howard A., "The Quenching of Solutions of Semilinear Hyperbolic Equations" (1981). *Mathematics Publications*. 45.

https://lib.dr.iastate.edu/math_pubs/45

## Comments

This is an article from

SIAM Journal on Mathematical Analysis12 (1981): 893, doi:10.1137/0512075. Posted with permission.