The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution. (English) Zbl 1540.35237
Summary: Let \(\mathcal{L}_{\Delta}\) 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
\[
\partial_t u + \mathcal{L}_{\Delta} u = 0 \text{ in } (0, \tfrac{N}{2}) \times \mathbb{R}^N, \qquad u(0, \cdot) = 0 \text{ in } \mathbb{R}^N \setminus \{0\}.
\]
We apply our results to give a classification of the solutions of
\[
\begin{cases}
\partial_t u + \mathcal{L}_{\Delta} u = 0 & \text{ in } (0, T) \times \mathbb{R}^N, \\
u (0, \cdot) = f & \text{ in } \mathbb{R}^N
\end{cases}
\]
and obtain an expression of the fundamental solution of the associated stationary equation in \(\mathbb{R}^N\), and of the fundamental solution \(u\) in a bounded domain, i.e. \(\mathcal{L}_{\Delta} u = k \delta_0\) in the sense of distributions in \(\Omega\), such that \(u = 0\) in \(\mathbb{R}^N \setminus \Omega\).
MSC:
| 35K15 | Initial value problems for second-order parabolic equations |
| 35A08 | Fundamental solutions to PDEs |
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