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From fast to very fast diffusion in the nonlinear heat equation


Author: Noureddine Igbida
Journal: Trans. Amer. Math. Soc. 361 (2009), 5089-5109
MSC (2000): Primary 35K60, 35K65, 35B40
DOI: https://doi.org/10.1090/S0002-9947-09-04540-1
Published electronically: May 6, 2009
MathSciNet review: 2515804
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Abstract: We study the asymptotic behavior of the sign-changing solution of the equation $ \displaystyle u_t=\nabla\cdot(\vert u\vert^{{-\alpha}} \nabla u)+f ,$ when the diffusion becomes very fast, i.e. as $ \displaystyle \alpha\uparrow 1.$ We prove that a solution $ u_\alpha(t)$ converges in $ \displaystyle L^1(\Omega ),$ uniformly for $ t$ in subsets with compact support in $ (0,T),$ to a solution of $ \displaystyle u_t=\nabla\cdot(\vert u\vert^{-1} \nabla u)+f .$ In contrast with the case of $ \alpha<1,$ we prove that the singularity 0 created in the limiting problem, i.e. $ \alpha=1,$ is an obstruction to the existence of sign-changing solutions. More precisely, we prove that, for each $ t\geq 0,$ the limiting solutions are either positive or negative or identically equal to 0 in all $ \Omega.$ This causes the limit to be singular, in the sense that a boundary layer appears at $ t=0,$ when one lets $ \alpha\uparrow 1.$


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Additional Information

Noureddine Igbida
Affiliation: LAMFA, CNRS-UMR 6140, Université de Picardie Jules Verne, 33 rue Saint Leu, 80038 Amiens, France
Email: noureddine.igbida@u-picardie.fr

DOI: https://doi.org/10.1090/S0002-9947-09-04540-1
Keywords: Singular limit, fast diffusion, logarithmic diffusion equation, degenerate parabolic equation, nonhomogeneous Neumann boundary condition, porous medium equation, sign-changing solution, boundary layer, semigroup of contraction.
Received by editor(s): February 4, 2005
Received by editor(s) in revised form: September 19, 2006, February 9, 2007, and March 12, 2007
Published electronically: May 6, 2009
Article copyright: © Copyright 2009 American Mathematical Society

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