تقرير
The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution
| العنوان: | The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution |
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| المؤلفون: | Chen, Huyuan, Véron, Laurent |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, 35K05, 35A08 |
| الوصف: | 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. $$\lnlap u=k\delta_0\quad {\rm in}\ \ \cD'(\Omega)\quad \text{such that }\,u=0\quad {\rm in}\ \ \R^N\setminus\Omega. $$ Comment: 55 pages. To appear in Journal of Functional Analysis |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2307.16197 |
| رقم الأكسشن: | edsarx.2307.16197 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |