Complete quenching for a degenerate parabolic problem with a localised nonlinear source
Summary: This paper studies a degenerate semilinear parabolic first initial boundary value problem with a nonlinear reaction \(f(u(b, t))\) taking place only at the single site \(b\) with \(\lim_{u\to c^-} f(u) = \infty\) for some positive constant \(c\). It is shown that there exists some \(t_q \leq \infty\) such that for \(0 \leq t < tq\), the problem has a unique nonnegative solution \(u\) before \(u(b, t)\) reaches \(c^-, u\) is a strictly increasing function of \(t\), and if \(t_q\)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 \(a \leq a^*\) while for \(a > a^*, u(b, t)\) 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.
scientific article; zbMATH DE number 6108262