Remarks on quenching

Abstract

Consider the parabolic problem

for $t>0,x\in {\mathbb{R}}^n$ under initial and boundary conditions $u=1$, say. Since $p$ is assumed positive, the right hand side becomes singular as $u\to 0$. When $u$ reaches zero in finite or infinite time, one says that the solution quenches in finite or infinite time. This article gives a survey of results on this kind of problem and emphasizes those that have been obtained at the SFB 123 in Heidelberg. It is an updated version of an invited survey lecture at the International Congress of Nonlinear Analysts in Tampa, August 1992. To be specific, I shall cover existence and nonexistence of quenching points, asymptotic behaviour of the solutions in space and time near the quenching points, qualitative behaviour, application to mean curvature flow and phase transitions, reaction in porous medium flow etc.

The tools are variational methods and suitable maximum principles. Many of the results presented in this article were obtained with my coauthors Acker, Dziuk, Fila, Kersner and Levine, but related results will also be mentioned.