The Quenching of Solutions of Some Nonlinear Parabolic Equations

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 $\lim _{u \to a} \phi (u) = \infty ,a 0$ 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)...