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

Let L-Delta be the logarithmic Laplacian operator with Fourier symbol 2 In |zeta|, we study the expression of the diffusion kernel which is associated to the equation partial derivative(t)u+L(Delta)u = 0 in (0,N/2)xR(N), u(0, .)=0 in R-N\{0). We apply our results to give a classification of the solutions of {partial derivative(t)u+L(Delta)u = 0 in (0, T) x 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 u in a bounded domain, i.e. L(Delta)u = k delta(0) in the sense of distributions in Omega, such that u = 0 in R-N\Omega. (c) 2024 Elsevier Inc. All rights reserved.