Abstract
SynopsisWe consider the diffusion-advection equation ut = uxx + (ε/(1 − u)β)x(ε >0, β >0), 0 < x < 1, t >0, under the boundary conditions ux + ε/(1 − u)β = 0. We prove that there is a critical number ε(β) such that when ε < ε(β) for certain initial data a global solution exists and converges to the corresponding stationary solution; any solution must quench (u reaches one in finite or infinite time) if ε ≧ε(β). We also show that quenching can only occur at x = 0, and that for each ε > 0 there exist initial data for which the solution quenches in finite time.
Publisher
Cambridge University Press (CUP)