Quenching for Semilinear Singular Parabolic Problems

Abstract

Let f be a real-valued function that is nondecreasing and continuously differentiable on $[0,c)$ for some finite $c(c > 0)$, satisfying the conditions $f(0) > 0$ and $\lim _{u \to c} f(u) = \infty $ . This article is concerned with positive solutions u of the semilinear singular parabolic differential equation $u_t = u_{xx} + ({b / x})u_x + f(u)$, $b < 1$, on a bounded interval $(0,a)$, which satisfy the initial condition $u(x,0) = 0$ and the boundary conditions $u(0,t) = 0$ and $u_x (a,t) = 0$. Let $| \cdot \|$ denote the sup-norm over the interval $[0,a]$. It is shown that a solution u quenches (i.e., there exists a $T < \infty $ such that $\lim _{t \to T,t < T} \| {u_t ( \cdot ,t)} \| = \infty )$ if $\| {u( \cdot ,t)} \|$ tends to c from below as t approaches T. Furthermore, there exists a critical length $a^ * $ such that u may exist for all $t > 0$ if $a < a^ * $, but $\| {u( \cdot ,t)} \|$ tends to c in finite time if $a > a^ * $. A numerical method is given to compute $a^ * $. An upper bound for the quenching time T is obtained. An example is given to illustrate the results.