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Abstract:

In this paper we study the blow-up problem for a non-local diffusion equation with a reaction term, ut (x, t) = ∫Ω J (x - y) (u (y, t) - u (x, t)) d y + up (x, t) . We prove that non-negative and non-trivial solutions blow up in finite time if and only if p > 1. Moreover, we find that the blow-up rate is the same as the one that holds for the ODE ut = up, that is, limt ↗ T (T - t)frac(1, p - 1) {norm of matrix} u ({dot operator}, t) {norm of matrix}∞ = (frac(1, p - 1))frac(1, p - 1). Next, we deal with the blow-up set. We prove single point blow-up for radially symmetric solutions with a single maximum at the origin, as well as the localization of the blow-up set near any prescribed point, for certain initial conditions in a general domain with p > 2. Finally, we show some numerical experiments which illustrate our results. © 2008 Elsevier Ltd. All rights reserved.

Registro:

Documento: Artículo
Título:Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term
Autor:Pérez-Llanos, M.; Rossi, J.D.
Filiación:Departamento de Matemáticas, U. Carlos III de Madrid, 28911 Leganés, Spain
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, U. de Buenos Aires, 1428 Buenos Aires, Argentina
Palabras clave:Blow-up; Non-local diffusion; Boundary conditions; Blow-up; Initial conditions; Neumann boundary condition; Nonlocal diffusion; Nontrivial solution; Numerical experiments; Radially symmetric solution; Single point blow-up; Diffusion
Año:2009
Volumen:70
Número:4
Página de inicio:1629
Página de fin:1640
DOI: http://dx.doi.org/10.1016/j.na.2008.02.076
Título revista:Nonlinear Analysis, Theory, Methods and Applications
Título revista abreviado:Nonlinear Anal Theory Methods Appl
ISSN:0362546X
CODEN:NOAND
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0362546X_v70_n4_p1629_PerezLlanos

Referencias:

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Citas:

---------- APA ----------
Pérez-Llanos, M. & Rossi, J.D. (2009) . Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term. Nonlinear Analysis, Theory, Methods and Applications, 70(4), 1629-1640.
http://dx.doi.org/10.1016/j.na.2008.02.076
---------- CHICAGO ----------
Pérez-Llanos, M., Rossi, J.D. "Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term" . Nonlinear Analysis, Theory, Methods and Applications 70, no. 4 (2009) : 1629-1640.
http://dx.doi.org/10.1016/j.na.2008.02.076
---------- MLA ----------
Pérez-Llanos, M., Rossi, J.D. "Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term" . Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 4, 2009, pp. 1629-1640.
http://dx.doi.org/10.1016/j.na.2008.02.076
---------- VANCOUVER ----------
Pérez-Llanos, M., Rossi, J.D. Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term. Nonlinear Anal Theory Methods Appl. 2009;70(4):1629-1640.
http://dx.doi.org/10.1016/j.na.2008.02.076