Math. Model. Nat. Phenom.
Volume 8, Number 3, 2013Front Propagation
|Page(s)||18 - 32|
|Published online||12 June 2013|
Asymptotic Behavior of Solutions to Diffusion Problems with Robin and Free Boundary Conditions
Department of Mathematics, Tongji
⋆ Corresponding author. E-mail: firstname.lastname@example.org
We study a nonlinear diffusion equation ut = uxx + f(u) with Robin boundary condition at x = 0 and with a free boundary condition at x = h(t), where h(t) > 0 is a moving boundary representing the expanding front in ecology models. For any f ∈ C1 with f(0) = 0, we prove that every bounded positive solution of this problem converges to a stationary one. As applications, we use this convergence result to study diffusion equations with monostable and combustion types of nonlinearities. We obtain dichotomy results and sharp thresholds for the asymptotic behavior of the solutions.
Mathematics Subject Classification: 35K20 / 35K55 / 35B40 / 35R35
Key words: Nonlinear diffusion equation / asymptotic behavior / Robin boundary condition / free boundary problem
© EDP Sciences, 2013
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