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On the existence and nonexistence of global solutions of a diffusion-advection equation

Published online by Cambridge University Press:  14 November 2011

Keng Deng
Keng Deng
Affiliation:
Department of Mathematics, University of Southwestern Louisiana, Lafayette, LA 70504-1010, U.S.A.
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Synopsis

We 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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 6 , 1993 , pp. 1099 - 1108
Copyright
Copyright © Royal Society of Edinburgh 1993

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