The Quenching of Solutions of Semilinear Hyperbolic Equations

We consider the problem $u_{tt} = u_{xx} + \phi (u(x,t)),0 0;u(0,t) = u(L,t) = 0;u(x,0) = u_t (x,0) = 0$. 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_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.