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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic behavior for a nonlocal diffusion equation with absorption and nonintegrable initial data. The supercritical case


Authors: Joana Terra and Noemi Wolanski
Journal: Proc. Amer. Math. Soc. 139 (2011), 1421-1432
MSC (2010): Primary 35K57, 35B40
Published electronically: September 2, 2010
MathSciNet review: 2748435
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Abstract: In this paper we study the asymptotic behavior as time goes to infinity of the solution to a nonlocal diffusion equation with absorption modeled by a powerlike reaction $ -u^p$, $ p>1$ and set in $ \mathbb{R}^N$. We consider a bounded, nonnegative initial datum $ u_0$ that behaves like a negative power at infinity. That is, $ \vert x\vert^\alpha u_0(x)\to A>0$ as $ \vert x\vert\to\infty$ with $ 0<\alpha\le N$. We prove that, in the supercritical case $ p>1+2/\alpha$, the solution behaves asymptotically as that of the heat equation (with diffusivity $ \mathfrak{a}$ related to the nonlocal operator) with the same initial datum.


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Joana Terra
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina
Email: jterra@dm.uba.ar

Noemi Wolanski
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina
Email: wolanski@dm.uba.ar

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10612-3
PII: S 0002-9939(2010)10612-3
Keywords: Nonlocal diffusion, boundary value problems.
Received by editor(s): November 25, 2009
Received by editor(s) in revised form: April 29, 2010
Published electronically: September 2, 2010
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2010 American Mathematical Society