Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term
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 blo...
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2009
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0362546X_v70_n4_p1629_PerezLlanos http://hdl.handle.net/20.500.12110/paper_0362546X_v70_n4_p1629_PerezLlanos |
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paper:paper_0362546X_v70_n4_p1629_PerezLlanos2023-06-08T15:35:21Z Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term Rossi, Julio Daniel 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 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. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2009 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0362546X_v70_n4_p1629_PerezLlanos http://hdl.handle.net/20.500.12110/paper_0362546X_v70_n4_p1629_PerezLlanos |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
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 |
| spellingShingle |
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 Rossi, Julio Daniel Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term |
| topic_facet |
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 |
| description |
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. |
| author |
Rossi, Julio Daniel |
| author_facet |
Rossi, Julio Daniel |
| author_sort |
Rossi, Julio Daniel |
| title |
Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term |
| title_short |
Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term |
| title_full |
Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term |
| title_fullStr |
Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term |
| title_full_unstemmed |
Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term |
| title_sort |
blow-up for a non-local diffusion problem with neumann boundary conditions and a reaction term |
| publishDate |
2009 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0362546X_v70_n4_p1629_PerezLlanos http://hdl.handle.net/20.500.12110/paper_0362546X_v70_n4_p1629_PerezLlanos |
| work_keys_str_mv |
AT rossijuliodaniel blowupforanonlocaldiffusionproblemwithneumannboundaryconditionsandareactionterm |
| _version_ |
1768542599767916544 |