The Quenching of Solutions of Some Nonlinear Parabolic Equations

Abstract

We consider the first initial-boundary value problem for $u_t = u_{xx} + \phi (u),\, 0 \leqq x \leqq l$ with $\phi > 0$ on $[0,a)$, $\phi $ convex, monotone increasing and $\lim _{u \to a} \phi (u) = \infty ,a < \infty $, and with $u(x,0) \equiv 0$. If $\Phi (c) = \smallint _0^c \phi (\eta )d\eta $, $\psi (c) = 2\sqrt 2 \int _0^{c{1 / 2}} {{dy} / \phi }(\Phi ^{ - 1} (c - y^2 ))$ and $l_0 = \sup \{ \Psi (c)\mid c \in ({\operatorname{Range }}\Phi ) \cap [0,\infty )\} $, we prove the following: (a) if $l < l_0 $,u exists for all $t > 0$ and approaches ($t \to \infty $), the smallest stationary solution of the differential equation; (b) if $l = l_0 $ and $l_0$ is taken by $\Psi $, then (a) holds; (c) if $l_0$ is not taken and ${\operatorname{Range}}\Phi $ is bounded, then u approaches from below the smallest weak stationary solution of the differential equation and this weak solution is not a strong stationary solution, $u_{xx} ({l / {2,t}}) \to - \infty $, and $u_t ({l / {2,t}}) \to 0$ as $t \to \infty $;(d) if $l = l_0 $ and Range $\Phi = [0,\infty )$ or (e) $l > l_0 $, then the existence interval is finite and $u({l / {2,t}}) \to a$ as $t \to T^ - $ for some $t < \infty $.