Article Dans Une Revue Journal of Functional Analysis Année : 2024

The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution

Résumé

Let L ∆ be the logarithmic Laplacian operator with Fourier symbol 2 ln |ζ|, we study the expression of the diffusion kernel which is associated to the equation ∂ t u + L ∆ u = 0 in (0, N 2) × R N , u(0, •) = 0 in R N \ {0}. We apply our results to give a classification of the solutions of ∂ t u + L ∆ u = 0 in (0, T) × R N u(0, •) = f in R N and obtain an expression of the fundamental solution of the associated stationary equation in R N , and of the fundamental solution in a bounded domain, i.e. L ∆ u = kδ 0 in D (Ω) such that u = 0 in R N \ Ω.

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Dates et versions

hal-04173733 , version 1 (30-07-2023)
hal-04173733 , version 2 (22-04-2024)

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Huyuan Chen, Laurent Véron. The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution. Journal of Functional Analysis, 2024, 287 (3), pp.110470. ⟨10.1016/j.jfa.2024.110470⟩. ⟨hal-04173733v2⟩
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