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Asymptotic Behavior of Solutions to Diffusion Problems withRobin and Free Boundary Conditions

Published online by Cambridge University Press:  12 June 2013

X. Liu  and
B. Lou
X. Liu
Affiliation:
Department of Mathematics, Tongji University, Shanghai 200092, China
B. Lou*
Affiliation:
Department of Mathematics, Tongji University, Shanghai 200092, China
*
⋆ Corresponding author. E-mail: [email protected]
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Abstract

We study a nonlinear diffusion equationut = uxx + f(u)with Robin boundary condition at x = 0 and with a free boundary conditionat x = h(t), whereh(t) > 0 is a moving boundary representing theexpanding front in ecology models. For anyf ∈ C1 with f(0) = 0, weprove that every bounded positive solution of this problem converges to a stationary one.As applications, we use this convergence result to study diffusion equations withmonostable and combustion types of nonlinearities. We obtain dichotomy results and sharpthresholds for the asymptotic behavior of the solutions.

Keywords

Nonlinear diffusion equationasymptotic behaviorRobin boundary conditionfree boundary problem
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 3: Front Propagation , 2013 , pp. 18 - 32
Copyright
© EDP Sciences, 2013

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