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

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 lim(u -> c)- f(u) = infinity for some positive constant c. It is shown that there exists some t(q) <= infinity such that for 0 <= t < t(q), the problem has a unique nonnegative solution u before u(b, t) reaches c-, u is a strictly increasing function of t, and if tq is finite, then u (b, t) reaches c-at t(q). The problem is shown to have a unique a* such that a unique global solution u exists for a <= a* while for a > a*, u (b, t) reaches c(-) at a finite tq. 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), ut blows up everywhere. A computational method is devised to compute the finite t(q). For illustration, an example is given.