Article Dans Une Revue Bulletin of the Belgian Mathematical Society - Simon Stevin Année : 2011

Singular behavior of the solution of the Cauchy-Dirichlet heat equation in weighted L^p -Sobolev spaces

Résumé

We consider the heat equation on a polygonal domain Ω of the plane in weighted L^p-Sobolev spaces ∂_t u − ∆u = h, in Ω × ]0, T [, u = 0, on ∂Ω × [0, T ], u(•, 0) = 0, in Ω. Here h belongs to L^p (0, T ; L^p_µ (Ω)), where L^p_µ (Ω) = {v ∈ L^p_loc (Ω) : r^µ v ∈ L^p (Ω)}, with a real parameter µ and r(x) the distance from x to the set of corners of Ω. We give sufficient conditions on µ, p and Ω that guarantee that trhe above problem has a unique solution u ∈ L^p (0, T ; L^p_µ (Ω)) that admits a decomposition into a regular part in weighted L^p-Sobolev spaces and an explicit singular part.

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Dates et versions

hal-03721680 , version 1 (12-07-2022)

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  • HAL Id : hal-03721680 , version 1

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Colette De Coster, Serge Nicaise. Singular behavior of the solution of the Cauchy-Dirichlet heat equation in weighted L^p -Sobolev spaces. Bulletin of the Belgian Mathematical Society - Simon Stevin, 2011, 18, pp.769-780. ⟨hal-03721680⟩
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