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

2023/07/30 by Huyuan Chen, Chen, Huyuan, Лаурент Верон +1 · 6 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2307.16197

Abstract

Let \lnlap 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+ \lnlap u=0 \rm in (0,\tfrac N2) × \R^N, u(0,⋅)=0 \rm in \R^N∖ \0\. We apply our results to give a classification of the solutions of \ \arraycolsep=1pt ∂_tu+ \lnlap u=0 · amp;\rm in (0,T)× \R^N
\phantom \lnlap u(0,⋅)=f · amp;\rmin \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. \lnlap u=kδ₀ \rm in \cD'(Ω) such that u=0 \rm in \R^N∖Ω.

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