On the quenching set for a fast diffusion equation: regional quenching

Abstract

We study positive solutions of a very fast diffusion equation, u_t = (u^m−1u_x)_x, m < 0, in a bounded interval, 0 < x < L, with a quenching-type boundary condition at one end, u (0, t) = (T − t)^{1/(1 − m)} and a zero-flux boundary condition at the other, (u^{m −1}u_x)(L, t) = 0. We prove that for m ≥ −1 regional quenching is not possible: the quenching set is either a single point or the whole interval. Conversely, if m < −1 single-point quenching is impossible, and quenching is either regional or global. For some lengths the above facts depend on the initial data. The results are obtained by studying the corresponding blow-up problem for the variable v = u^{m −1}.