From fast to very fast diffusion in the nonlinear heat equation
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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 $\alpha \uparrow 1.$References
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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
- 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
- © Copyright 2009 American Mathematical Society
- 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
- MathSciNet review: 2515804