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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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_0362546X_v70_n4_p1629_PerezLlanos |
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| Sumario: | 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. |
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