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

Abstract

Let LΔ be the logarithmic Laplacian operator with Fourier symbol 2ln⁡|ζ|, we study the expression of the diffusion kernel which is associated to the equation∂tu+LΔu=0in(0,N2)×RN,u(0,⋅)=0inRN∖{0}. We apply our results to give a classification of the solutions of{∂tu+LΔu=0in(0,T)×RN,u(0,⋅)=finRN and obtain an expression of the fundamental solution of the associated stationary equation in RN, and of the fundamental solution u in a bounded domain, i.e. LΔu=kδ0 in the sense of distributions in Ω, such that u=0 in RN∖Ω.