Global Solutions for a Nonlocal Problem with Logarithmic Source Term
The current paper discusses the global existence and asymptotic behavior of solutions of the following new nonlocal problem$$ u_{tt}- M\left(\displaystyle \int_{\Omega}|\nabla u|^{2}\, dx\right)\triangle u + \delta u_{t}= |u|^{\rho-2}u \log|u|, \quad \text{in}\ \Omega \times ]0,\infty[, $$where\beg...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
University of Maragheh
2024-07-01
|
| Series: | Sahand Communications in Mathematical Analysis |
| Subjects: | |
| Online Access: | https://scma.maragheh.ac.ir/article_712733_621d736ee4c56b9ae3237a0bb849f3d1.pdf |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | The current paper discusses the global existence and asymptotic behavior of solutions of the following new nonlocal problem$$ u_{tt}- M\left(\displaystyle \int_{\Omega}|\nabla u|^{2}\, dx\right)\triangle u + \delta u_{t}= |u|^{\rho-2}u \log|u|, \quad \text{in}\ \Omega \times ]0,\infty[, $$where\begin{equation*}M(s)=\left\{\begin{array}{ll}{a-bs,}&{\text{for}\ s \in [0,\frac{a}{b}[,}\\{0,}&{\text{for}\ s \in [\frac{a}{b}, +\infty[.}\end{array}\right.\end{equation*}If the initial data are appropriately small, we derive existence of global strong solutions and the exponential decay of the energy. |
|---|---|
| ISSN: | 2322-5807 2423-3900 |