Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-2010-10612-3
Published electronically: September 2, 2010
MathSciNet review: 2748435
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. P. Bates, A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher dimensions, J. Statistical Phys. 95, 1999, 1119-1139. MR 1712445 (2000j:82020)
  • 2. P. Bates, A. Chmaj, A discrete convolution model for phase transitions, Arch. Rat. Mech. Anal. 150, 1999, 281-305. MR 1741258 (2001c:82026)
  • 3. P. Bates, P. Fife, X. Ren, X. Wang, Travelling waves in a convolution model for phase transitions, Arch. Rat. Mech. Anal. 138, 1997, 105-136. MR 1463804 (98f:45004)
  • 4. C. Carrillo, P. Fife, Spatial effects in discrete generation population models, J. Math. Biol. 50(2), 2005, 161-188. MR 2120547 (2005k:92048)
  • 5. M. Chaves, E. Chasseigne, J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. (9) 86(3), 2006), 271-291. MR 2257732 (2007e:35279)
  • 6. C. Cortazar, M. Elgueta, F. Quiros, N. Wolanski, Large time behavior of the solution to the Dirichlet problem for a nonlocal diffusion equation in an exterior domain, in preparation.
  • 7. C. Cortazar, M. Elgueta, J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions, Israel Journal of Mathematics 170(1), 2009, 53-60. MR 2506317 (2010e:35197)
  • 8. C. Cortazar, M. Elgueta, J. D. Rossi, N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Rat. Mech. Anal. 187(1), 2008, 137-156. MR 2358337 (2008k:35261)
  • 9. P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, 153-191, Springer, Berlin, 2003. MR 1999098 (2004h:35100)
  • 10. G. Gilboa, S. Osher, Nonlocal operators with application to image processing, Multiscale Model. Simul. 7(3), 2008, 1005-1028. MR 2480109 (2010b:94006)
  • 11. L. Grafakos, Classical Fourier analysis. Second edition. Graduate Texts in Mathematics, 249. Springer, New York, 2008. MR 2445437
  • 12. L. Herraiz, Asymptotic behaviour of solutions of some semilinear parabolic problems, Ann. Inst. Henri Poincaré 16(1), 1999, 49-105. MR 1668560 (2000a:35106)
  • 13. L. I. Ignat, J. D. Rossi, Refined asymptotic expansions for nonlocal diffusion equations, J. Evolution Equations. 8, 2008, 617-629. MR 2460931 (2009j:45001)
  • 14. S. Kamin, L. A. Peletier, Large time behavior of solutions of the heat equation with absorption, Anal. Scuola. Norm. Sup. Pisa Serie 4, 12, 1985, 393-408. MR 837255 (87h:35140)
  • 15. S. Kamin, M. Ughi, On the behaviour as $ t\to\infty$ of the solutions of the Cauchy problem for certain nonlinear parabolic equations, J. Math. Anal. Appl. 128, 1997, 456-469. MR 917378 (89m:35029)
  • 16. C. Lederman, N. Wolanski, Singular perturbation in a nonlocal diffusion model, Communications in PDE 31(2), 2006, 195-241. MR 2209752 (2007e:35166)
  • 17. A. Pazoto, J. D. Rossi, Asymptotic behaviour for a semilinear nonlocal equation. Asymptotic Analysis 52(1-2), 2007, 143-155. MR 2337030 (2008h:35023)
  • 18. J. Terra, N. Wolanski, Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data, submitted.
  • 19. L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, J. Differential Equations 197(1), 2004, 162-196. MR 2030153 (2004m:45010)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35K57, 35B40

Retrieve articles in all journals with MSC (2010): 35K57, 35B40


Additional Information

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: https://doi.org/10.1090/S0002-9939-2010-10612-3
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

American Mathematical Society