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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