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

Huyuan Chen (1) , Laurent Véron (2)

1 Jiangxi Normal University
2 IDP - Institut Denis Poisson

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.

Mots clés: Cauchy problem; logarithmic Laplacian; Fundamental solution

Domaines: Equations aux dérivées partielles [math.AP]

Huyuan Chen, Laurent Véron. The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution. 2023. ⟨hal-04173733v1⟩