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Quenching phenomena for a non-local diffusion equation with a singular absorption

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Abstract

In this paper we study the quenching problem for the non-local diffusion equation

$${u_t}(x,t) = \int\limits_\Omega {J(x - y)u(y,t)dy + \int\limits_{{\mathbb{R}^N}\backslash \Omega } {J(x - y)dy - u(x,t) - \lambda {u^{ - p}}(x,t)} } $$

. We prove that there exists a critical parameter λ * such that for all λ > λ * every solution quenches and for λ ≤ λ * there are both global and quenching solutions. For the quenching solutions we study the quenching rate and the quenching set. We also prove that the solutions of properly rescaled non local problems approximate the solution of the semilinear heat equation with u = 1 at the boundary.

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  1. Departamento de Matemáticas, U. Complutense de Madrid, 28040, Madrid, Spain

    Raúl Ferreira

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  1. Raúl Ferreira
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Correspondence to Raúl Ferreira.

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Ferreira, R. Quenching phenomena for a non-local diffusion equation with a singular absorption. Isr. J. Math. 184, 387–402 (2011). https://doi.org/10.1007/s11856-011-0072-y

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  • DOI: https://doi.org/10.1007/s11856-011-0072-y

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