Decay of mass for fractional evolution equation with memory term
Authors:
Ahmad Z. Fino, Hassan Ibrahim and Bilal Barakeh
Journal:
Quart. Appl. Math. 71 (2013), 215-228
MSC (2010):
Primary 35K55, 35B40
DOI:
https://doi.org/10.1090/S0033-569X-2012-01286-4
Published electronically:
August 27, 2012
MathSciNet review:
3087420
Full-text PDF Free Access
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Abstract: We investigate the decay properties of the mass $M(t)= \int _{\mathbb {R}^N} u(\cdot ,t)dx$ of the solutions of a fractional diffusion equation with nonlinear memory term. We show, using a suitable class of initial data and a restriction on the diffusion and nonlinear term, that the memory term determines the large time asymptotics; that is, $M(t)$ tends to zero as $t\to \infty .$
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References
- M. Ben-Artzi, H. Koch, Decay of mass for a semilinear parabolic equation, Comm. Partial Differential Equations 24 (1999), 869–881. MR 1680909 (2000a:35098)
- P. Biler, G. Karch, W. A. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. Henri Poincaré Analyse Non Linéaire, 18 (2001), 613–637. MR 1849690 (2002f:35035)
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- A. Z. Fino, M. Kirane, Qualitative properties of solutions to a time-space fractional evolution equation, Quarterly of Applied Mathematics 70 (2012), 133–157.
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- F. Weissler, Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), 29–40. MR 599472 (82g:35059)
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Additional Information
Ahmad Z. Fino
Affiliation:
LaMA-Liban, Lebanese University, P.O. Box 37 Tripoli, Lebanon, and School of Arts and Sciences, Lebanese International University (LIU), Tripoli Campus, Dahr el Ain Road, Tripoli, Lebanon
Email:
ahmad.fino01@gmail.com, afino@ul.edu.lb
Hassan Ibrahim
Affiliation:
Lebanese University, Faculty of Sciences-I, Hadath, Beirut, Lebanon & LaMA-Liban, Lebanese University, P.O. Box 37 Tripoli, Lebanon, and School of Arts and Sciences, Lebanese International University (LIU), Beirut Campus, Al-Mouseitbeh, P.B. Box 14-6404, Beirut, Lebanon
Email:
ibrahim@cermics.enpc.fr
Bilal Barakeh
Affiliation:
LaMA-Liban, Lebanese University, P.O. Box 37 Tripoli, Lebanon
Email:
bilal.barakeh@hotmail.com
Keywords:
Large time behavior of solutions,
semilinear parabolic equation,
fractional Laplacian
Received by editor(s):
May 17, 2011
Published electronically:
August 27, 2012
Article copyright:
© Copyright 2012
Brown University