The Quenching of Solutions of Linear Parabolic and Hyperbolic Equations with Nonlinear Boundary Conditions

In this paper we examine the initial-boundary value problems $(\alpha ):u_t = u_{xx} $, $0 < x < L,t > 0$, $u(0,t) = u(x,0) = 0$, $u_x (L,t) = \phi (u(L,t))$ and $(\beta ):u_{tt} = u_{xx} $, $0 < x < L,t > 0$, $u(0,t) = u(x,0) = u_t (x,0)$, $u_x (L,t) = \phi (u(L,t))$ where $\phi ( - \infty ,1) \to (0,\infty )$ is continuously differentiable, monotone increasing and $\lim _{u \to 1} - \phi (u) = + \infty $. For problem $(\alpha )$ we show that there is a positive number $L_0 $ such that if $L \leq L_0 $, $u(x,t) \leq 1 - \delta $ for some $\delta > 0$ for all $t > 0$, while if $L > L_0 ,u(L,t)$ reaches one in finite time while $u_t (L,t)$ becomes unbounded in that time. For problem $(\beta )$ it is shown that if L is sufficiently small, then $u(L,t) \leq 1 - \delta $ for all $t > 0$ while if L is sufficiently large and $\int_0^1 \phi (\eta )d\eta < \infty ,u(L,t)$ reaches one in finite time whereas if $\int_0^1 \phi (\eta )d\eta = \infty ,u(L,t)$ reaches one in finite or infinite time. In either of the last two situations $u_t (L,t)$ becomes unbounded if the time interval is finite. If u reaches one in infinite time, then $\int_0^1 {u_x^2 } (x,t)dx$ and $u(x,t)$ are unbounded on the half line and half strip respectively.