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

Let $\text{lnlap}$ be the logarithmic Laplacian operator with Fourier symbol $2\ln \mathrm{\mid }\zeta \mathrm{\mid }$, we study the expression of the diffusion kernel which is associated to the equation $${\mathrm{\partial }}_{t}u+\text{lnlap}u=0\mathrm{i}\mathrm{n}(0,\frac{N}{2})\times {\mathbb{R}}^N,u(0,\cdot )=0\mathrm{i}\mathrm{n}{\mathbb{R}}^N\setminus \{0\}\mathrm{.}$$ We apply our results to give a classification of the solutions of $$\{\text{arraycolsep}=1pt\begin{array}{ll}{\displaystyle {\mathrm{\partial }}_{t}u+\text{lnlap}u=0}& \mathrm{i}\mathrm{n}(0,T)\times {\mathbb{R}}^N\\ \phantom{\text{lnlap}}{\displaystyle u(0,\cdot )=f}& \mathrm{i}\mathrm{n}{\mathbb{R}}^N\end{array}$$ and obtain an expression of the fundamental solution of the associated stationary equation in ${\mathbb{R}}^N$, and of the fundamental solution in a bounded domain, i.e. $$\text{lnlap}u=k{\delta }_0\mathrm{i}\mathrm{n}{\text{cD}}^{\mathrm{\prime }}(\mathrm{\Omega })\text{such that }u=0\mathrm{i}\mathrm{n}{\mathbb{R}}^N\setminus \mathrm{\Omega }\mathrm{.}$$