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

TL;DR

This paper investigates the diffusion kernel of the logarithmic Laplacian operator, classifies solutions to related PDEs, and derives fundamental solutions in both unbounded and bounded domains.

Contribution

It provides explicit expressions for the diffusion kernel, classifies solutions of the associated PDE, and characterizes fundamental solutions for the logarithmic Laplacian.

Findings

Derived the diffusion kernel for the logarithmic Laplacian.

Classified solutions to the PDE with initial data.

Explicit formulas for fundamental solutions in various domains.

Abstract

Let $\lnlap$ be the logarithmic Laplacian operator with Fourier symbol $2\ln |\zeta|$, we study the expression of the diffusion kernel which is associated to the equation $$\partial_tu+ \lnlap u=0 \ \ {\rm in}\ \, (0,\tfrac N2) \times \R^N,\quad\quad u(0,\cdot)=0\ \ {\rm in}\ \, \R^N\setminus \{0\}.$$ We apply our results to give a classification of the solutions of $$\left\{ \arraycolsep=1pt \begin{array}{lll} \displaystyle \partial_tu+ \lnlap u=0\quad \ &{\rm in}\ \ (0,T)\times \R^N\\[2.5mm] \phantom{ \lnlap \ \, } \displaystyle u(0,\cdot)=f\quad \ &{\rm{in}}\ \ \R^N \end{array} \right. $$ 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. $$\lnlap u=k\delta_0\quad {\rm in}\ \ \cD'(\Omega)\quad \text{such that }\,u=0\quad {\rm in}\ \ \R^N\setminus\Omega. $$