Complete quenching for a degenerate parabolic problem with a localised nonlinear source

Abstract

This paper studies a degenerate semilinear parabolic first initialboundary value problem with a nonlinear reaction f ( u ( b , t )) taking place only at the single site b with limu →c − f ( u ) = ∞ for some positive constant c . It is shown that there exists some t q ≤∞ such that for 0 ≤ t a *, u ( b , t ) reaches c− at a finite t q . A formula relating a *, b and f is given, and no quenching in infinite time is deduced. It is also shown that when u ( b , t ) reaches c at a finite t q , u t blows up everywhere. A computational method is devised to compute the finite t q . For illustration, an example is given.