Semilinear heat equation with singular terms
The main goal of this paper is to analyze the existence and nonexistence as well as the regularity of positive solutions for the following initial parabolic problem $$ \begin{cases} \partial_tu-\Delta u=\displaystyle\mu\frac{u}{|x|^{2}}+\frac{f}{u^{\sigma}}&\mbox{in } \Omega_T:=\Omega\times(0,...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2022-12-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=9983 |
| _version_ | 1797830459034435584 |
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| author | Mohamed Mahmoud Ould Khatri Ahmed Youssfi |
| author_facet | Mohamed Mahmoud Ould Khatri Ahmed Youssfi |
| author_sort | Mohamed Mahmoud Ould Khatri |
| collection | DOAJ |
| description | The main goal of this paper is to analyze the existence and nonexistence as well as the regularity of positive solutions for the following initial parabolic problem
$$
\begin{cases}
\partial_tu-\Delta u=\displaystyle\mu\frac{u}{|x|^{2}}+\frac{f}{u^{\sigma}}&\mbox{in } \Omega_T:=\Omega\times(0,T),\\
u=0&\mbox{on } \partial\Omega\times(0,T),\\
u(x,0)=u_{0}(x)&\mbox{in } \Omega,
\end{cases}
$$
where $\Omega\subset\mathbb{R}^N$, $N\geq3$, is a bounded open, $\sigma\geq0$ and $\mu>0$ are real constants and $f\in L^m(\Omega_T)$, $m\geq1$, and $u_0$ are nonnegative functions. The study we lead shows that the existence of solutions depends on $\sigma$ and the summability of the datum $f$ as well as on the interplay between $\mu$ and the best constant in the Hardy inequality. Regularity results of solutions, when they exist, are also provided. Furthermore, we prove uniqueness of finite energy solutions. |
| first_indexed | 2024-04-09T13:36:23Z |
| format | Article |
| id | doaj.art-af5403481abb414ab9bf5870ccf0f474 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:36:23Z |
| publishDate | 2022-12-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-af5403481abb414ab9bf5870ccf0f4742023-05-09T07:53:12ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752022-12-0120226913410.14232/ejqtde.2022.1.699983Semilinear heat equation with singular termsMohamed Mahmoud Ould Khatri0Ahmed Youssfi1Sidi Mohamed Ben Abdellah University, National School of Applied Sciences, P.O. Box 72 Fès-Principale, Fez, 30 000, MoroccoSidi Mohamed Ben Abdellah University, Laboratory of Mathematical Analysis and Applications-FSDM, National School of Applied Sciences, P.O. Box 72 Fès-Principale, Fez, 30 000, MoroccoThe main goal of this paper is to analyze the existence and nonexistence as well as the regularity of positive solutions for the following initial parabolic problem $$ \begin{cases} \partial_tu-\Delta u=\displaystyle\mu\frac{u}{|x|^{2}}+\frac{f}{u^{\sigma}}&\mbox{in } \Omega_T:=\Omega\times(0,T),\\ u=0&\mbox{on } \partial\Omega\times(0,T),\\ u(x,0)=u_{0}(x)&\mbox{in } \Omega, \end{cases} $$ where $\Omega\subset\mathbb{R}^N$, $N\geq3$, is a bounded open, $\sigma\geq0$ and $\mu>0$ are real constants and $f\in L^m(\Omega_T)$, $m\geq1$, and $u_0$ are nonnegative functions. The study we lead shows that the existence of solutions depends on $\sigma$ and the summability of the datum $f$ as well as on the interplay between $\mu$ and the best constant in the Hardy inequality. Regularity results of solutions, when they exist, are also provided. Furthermore, we prove uniqueness of finite energy solutions.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=9983heat equationexistence and regularityhardy potentialsingular terms |
| spellingShingle | Mohamed Mahmoud Ould Khatri Ahmed Youssfi Semilinear heat equation with singular terms Electronic Journal of Qualitative Theory of Differential Equations heat equation existence and regularity hardy potential singular terms |
| title | Semilinear heat equation with singular terms |
| title_full | Semilinear heat equation with singular terms |
| title_fullStr | Semilinear heat equation with singular terms |
| title_full_unstemmed | Semilinear heat equation with singular terms |
| title_short | Semilinear heat equation with singular terms |
| title_sort | semilinear heat equation with singular terms |
| topic | heat equation existence and regularity hardy potential singular terms |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=9983 |
| work_keys_str_mv | AT mohamedmahmoudouldkhatri semilinearheatequationwithsingularterms AT ahmedyoussfi semilinearheatequationwithsingularterms |