Quenching for A Nonlinear Diffusion Equation with Singular Boundary Outfluxes

Abstract

In this paper, we study a nonlinear diffusion equation (φ(u))_{t}=u_{xx}, 0<x<a, t>0 with singular boundary outfluxes u_{x}(0,t)=u^{-p}(0,t), u_{x}(a,t)=-u^{-q}(a,t). Firstly, we get the quenchnig occurs in a finite time at the boundary x=a under certain conditions. Finally, we show the time derivative blows up at the quenching time and we also establish results on quenching time and rate for certain nonlinearities.