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Abstract:

We study the large time behavior of nonnegative solutions of the Cauchy problem ut =R J(x,y)(u(y; t),u(x; t)) dy,up, u(x; 0) = u0(x) 2 L∞, where |x|αu0(x) → A < 0 as |x| → 1. One of our main goals is the study of the critical case p = 1+2=ff for 0 > ff > N, left open in previous articles, for which we prove that tff=2ju(x; t) , U(x; t)j → 0 where U is the solution of the heat equation with absorption with initial datum U(x; 0) = CA;N |x|,α. Our proof, involving sequences of rescalings of the solution, allows us to establish also the large time behavior of solutions having more general nonintegrable initial data u0 in the supercritical case and also in the critical case (p = 1 + 2=N) for bounded and integrable u0.

Registro:

Documento: Artículo
Título:Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data
Autor:Terra, J.; Wolanski, N.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Palabras clave:Large time behavior; Nonlocal diffusion
Año:2011
Volumen:31
Número:2
Página de inicio:581
Página de fin:605
DOI: http://dx.doi.org/10.3934/dcds.2011.31.581
Handle:http://hdl.handle.net/20.500.12110/paper_10780947_v31_n2_p581_Terra
Título revista:Discrete and Continuous Dynamical Systems
Título revista abreviado:Discrete Contin. Dyn. Syst.
ISSN:10780947
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_10780947_v31_n2_p581_Terra.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10780947_v31_n2_p581_Terra

Referencias:

  • Bates, P., Chmaj, A., An integrodifferential model for phase transitions: Stationary solu- tions in higher dimensions (1999) J. Statistical Phys., 95, pp. 1119-1139
  • Bates, P., Chmaj, A., A discrete convolution model for phase transitions (1999) Arch. Rat. Mech. Anal., 150, pp. 281-305
  • Bates, P., Fife, P., Ren, X., Wang, X., Travelling waves in a convolution model for phase transitions (1997) Arch. Rat. Mech. Anal., 138, pp. 105-136
  • Bates, P., Zhao, G., Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal (2007) J. Math. Anal. Appl., 332, pp. 428-440
  • Carrillo, C., Fife, P., Spatial effects in discrete generation population models (2005) Journal of Mathematical Biology, 50 (2), pp. 161-188. , DOI 10.1007/s00285-004-0284-4
  • Chasseigne, E., Chaves, M., Rossi, J.D., Asymptotic behavior for nonlocal diffusion equations (2006) Journal des Mathematiques Pures et Appliquees, 86 (3), pp. 271-291. , DOI 10.1016/j.matpur.2006.04.005, PII S0021782406000559
  • Chen, X., Qi, Y.W., Wang, M., Long time behavior of solutions to p-laplacian equation with absorption (2003) SIAM Jour. Math. Anal., 35, pp. 123-134
  • Cortazar, C., Elgueta, M., Quiros, F., Wolanski, N., Large Time Behavior of the Solution to the Dirichlet Problem for a Nonlocal Diffusion Equation in an Exterior Domain, , in preparation
  • Cortazar, C., Elgueta, M., Rossi, J.D., Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions (2009) Israel Journal of Mathematics., 170, pp. 53-60
  • Cortazar, C., Elgueta, M., Rossi, J.D., Wolanski, N., How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems (2008) Arch. Rat. Mech. Anal., 187, pp. 137-156
  • Fife, P., Some nonclassical trends in parabolic and parabolic-like evolutions (2003) Trends in Non- linear Analysis, pp. 153-191. , Springer, Berlin
  • Gilboa, G., Osher, S., Nonlocal operators with application to image processing (2008) Multiscale Model. Simul., 7, pp. 1005-1028
  • Grafakos, L., (2004) Classical and Modern Fourier Analysis, , Pearson Education, Inc., Upper Saddle River, NJ
  • Herraiz, L., Asymptotic behaviour of solutions of some semilinear parabolic problems (1999) Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis, 16 (1), pp. 49-105
  • Ignat, L.I., Rossi, J.D., Refined asymptotic expansions for nonlocal diffusion equations (2008) J. Evolution Equations, 8, pp. 617-629
  • Kamin, S., Peletier, L.A., Large time behavior of solutions of the heat equation with absorption (1985) Anal. Scuola. Norm. Sup. Pisa Serie, 4 (12), pp. 393-408
  • Kamin, S., Peletier, L.A., Large time behavior of solutions of the porous media equation with absorption (1986) Israel J. Math., 55, pp. 129-146
  • Kamin, S., Ughi, M., On the behavior as t rarr; 1 of the solutions of the Cauchy problem for certain nonlinear parabolic equations (1987) J. Math. Anal. Appl., 128, pp. 456-469
  • Lederman, C., Wolanski, N., Singular perturbation in a nonlocal diffusion problem (2006) Communications in Partial Differential Equations, 31 (2), pp. 195-241. , DOI 10.1080/03605300500358111, PII M705432860742
  • Pazoto, A.F., Rossi, J.D., Asymptotic behaviour for a semilinear nonlocal equation (2007) Asymptotic Analysis, 52 (1-2), pp. 143-155
  • Terra, J., Wolanski, N., Asymptotic behavior for a nonlocal diffusion equation with ab- sorption and nonintegrable initial data. The supercritical case (2011) Proc. Amer. Math. Soc., 139, pp. 1421-1432
  • Zhang, L., Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks (2004) Journal of Differential Equations, 197 (1), pp. 162-196. , DOI 10.1016/S0022-0396(03)00170-0
  • Zhao, J., The Asymptotic Behavior of solutions of a quasilinear degenerate parabolic equation (1993) J. Differential Equations, 102, pp. 33-52

Citas:

---------- APA ----------
Terra, J. & Wolanski, N. (2011) . Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete and Continuous Dynamical Systems, 31(2), 581-605.
http://dx.doi.org/10.3934/dcds.2011.31.581
---------- CHICAGO ----------
Terra, J., Wolanski, N. "Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data" . Discrete and Continuous Dynamical Systems 31, no. 2 (2011) : 581-605.
http://dx.doi.org/10.3934/dcds.2011.31.581
---------- MLA ----------
Terra, J., Wolanski, N. "Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data" . Discrete and Continuous Dynamical Systems, vol. 31, no. 2, 2011, pp. 581-605.
http://dx.doi.org/10.3934/dcds.2011.31.581
---------- VANCOUVER ----------
Terra, J., Wolanski, N. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete Contin. Dyn. Syst. 2011;31(2):581-605.
http://dx.doi.org/10.3934/dcds.2011.31.581