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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2307.16197 |
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| _version_ | 1866911850080239616 |
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| author | Chen, Huyuan Véron, Laurent |
| author_facet | Chen, Huyuan Véron, Laurent |
| contents | Let $\lnlap$ be the logarithmic Laplacian operator with Fourier symbol $2\ln |ζ|$, 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. $$\lnlap u=kδ_0\quad {\rm in}\ \ \cD'(Ω)\quad \text{such that }\,u=0\quad {\rm in}\ \ \R^N\setminusΩ. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_16197 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution Chen, Huyuan Véron, Laurent Analysis of PDEs 35K05, 35A08 Let $\lnlap$ be the logarithmic Laplacian operator with Fourier symbol $2\ln |ζ|$, 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. $$\lnlap u=kδ_0\quad {\rm in}\ \ \cD'(Ω)\quad \text{such that }\,u=0\quad {\rm in}\ \ \R^N\setminusΩ. $$ |
| title | The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution |
| topic | Analysis of PDEs 35K05, 35A08 |
| url | https://arxiv.org/abs/2307.16197 |