Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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
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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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  • 1. 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)
  • 2. 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)
  • 3. K. Bogdan, T. Byczkowski, Potential theory for the $ \alpha $-stable Schrödinger operator on bounded Lipschitz domains, Studia Mathematica 133 (1999), no. 1, 53-92. MR 1671973 (99m:31010)
  • 4. T. Cazenave, F. Dickstein, F. D. Weissler, An equation whose Fujita critical exponent is not given by scaling, Nonlinear Analysis 68 (2008), 862-874. MR 2382302 (2009c:35218)
  • 5. T. Cazenave, A. Haraux, Introduction aux problèmes d'évolution semi-linéaires, Ellipses, Paris, (1990). MR 1299976 (95f:35002)
  • 6. J. Droniou, C. Imbert, Fractal first-order partial differential equations, Arch. Rational Mech. Anal. 182 (2006), 299-331. MR 2259335 (2009c:35037)
  • 7. A. Z. Fino, V. Georgiev, M. Kirane, Finite time blow-up for a wave equation with a nonlocal nonlinearity, submitted, arXiv:1008.4219.
  • 8. A. Fino, G. Karch, Decay of mass for nonlinear equation with fractional Laplacian, J. Monatsh. Math. 160 (2010), 375-384. MR 2661320
  • 9. A. Z. Fino, M. Kirane, Qualitative properties of solutions to a time-space fractional evolution equation, Quarterly of Applied Mathematics 70 (2012), 133-157.
  • 10. H. Fujita, On the blowing up of solutions of the problem for $ u_t=\Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo 13 (1966), 109-124. MR 0214914 (35:5761)
  • 11. M. Guedda, M. Kirane, Criticality for some evolution equations, Differential Equations 37 (2001), 511-520. MR 1854046 (2002h:35028)
  • 12. N. Ju, The Maximum Principle and the Global Attractor for the Dissipative $ 2$-D Quasi-Geostrophic Equations, Comm. Pure. Appl. Ana. 255 (2005), 161-181. MR 2123380 (2005m:37194)
  • 13. G. Karch, W.A. Woyczyński, Fractal Hamilton-Jacobi-KPZ equations, Trans. Amer. Math. Soc. 360 (2008), 2423-2442. MR 2373320 (2008m:35182)
  • 14. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies), 2006. MR 2218073 (2007a:34002)
  • 15. M. Kirane, M. Qafsaoui, Global nonexistence for the Cauchy problem of some nonlinear reaction-diffusion systems, J. Math. Analysis and Appl. 268 (2002), 217-243. MR 1893203 (2004a:35123)
  • 16. D. Lamberton, Equations d'évolution linéaires associées à des semi-groupes de contractions dans les espaces $ L^p$, J. Functional Analysis, 72 (1987), 252-262. MR 886813 (88g:47085)
  • 17. Ph. Laurençot, Ph. Souplet, On the growth of mass for a viscous Hamilton-Jacobi equation, J. Anal. Math. 89 (2003), 367-383. MR 1981925 (2004c:35188)
  • 18. F. Mainardi, Fractional Calculus andWaves in Linear Viscoelasticity, Imperial College Press, London, UK, 2010. MR 2676137 (2011e:74002)
  • 19. E. Mitidieri, S. I. Pohozaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov. Inst. Math. 234 (2001), 1-383. MR 1879326 (2005d:35004)
  • 20. M. Nagasawa, T. Sirao, Probabilistic treatment of the blowing up of solutions for a nonlinear integral equation, Trans. Amer. Math. Soc. 139 (1969), 301-310. MR 0239379 (39:736)
  • 21. Jace W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187-204. MR 0295683 (45:4749)
  • 22. R. G. Pinsky, Decay of mass for the equation $ u_t=\Delta u- a(x)u^p\vert{\nabla u}\vert^q$, J. Diff. Eq. 165 (2000), 1-23. MR 1771786 (2001g:35109)
  • 23. C. A. Roberts, W. E. Olmstead, Blow-up in a subdiffusive medium of infinite extent, Fract. Calc. Appl. Anal. 12 (2009), 179-194. MR 2498365 (2010d:35381)
  • 24. S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math. 12 (1975), 45-51. MR 0470493 (57:10247)
  • 25. K. Yosida, ``Functional Analysis'', sixth edition, Springer-Verlag, Berlin, Heidelberg, New York, 1980. MR 617913 (82i:46002)
  • 26. Qi S. Zhang, A blow up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris 333 (2001), 109-114. MR 1847355 (2003d:35189)
  • 27. Wei Zhang, Nobuyuki Shimizu, Damping properties of the viscoelastic material described by fractional Kelvin-Voigt model, JSME international journal. Series C, Mechanical systems, machine elements and manufacturing, 42 (1999), no. 1, 1-9.
  • 28. 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)
  • 29. F. Weissler, Semilinear Evolution Equations in Banach Spaces, J. Functional Analysis 32 (1979), 277-296. MR 538855 (80i:47091)

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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

DOI: https://doi.org/10.1090/S0033-569X-2012-01286-4
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

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