## Mathematics Publications

Mathematics

Article

#### Publication Version

Published Version

1981

#### Journal or Book Title

SIAM Journal on Mathematical Analysis

12

6

893

903

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.

This is an article from SIAM Journal on Mathematical Analysis 12 (1981): 893, doi:10.1137/0512075. Posted with permission.

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