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...
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| Language: | English |
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University of Maragheh
2024-07-01
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| Series: | Sahand Communications in Mathematical Analysis |
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| Online Access: | https://scma.maragheh.ac.ir/article_712733_621d736ee4c56b9ae3237a0bb849f3d1.pdf |
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| _version_ | 1823859399487127552 |
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| author | Eugenio Lapa |
| author_facet | Eugenio Lapa |
| author_sort | Eugenio Lapa |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-69a8a47c00c44224a16a77e801e4aea3 |
| institution | Kabale University |
| issn | 2322-5807 2423-3900 |
| language | English |
| publishDate | 2024-07-01 |
| publisher | University of Maragheh |
| record_format | Article |
| series | Sahand Communications in Mathematical Analysis |
| spelling | doaj-art-69a8a47c00c44224a16a77e801e4aea32025-02-11T05:27:31ZengUniversity of MaraghehSahand Communications in Mathematical Analysis2322-58072423-39002024-07-0121337138510.22130/scma.2023.2001016.1307712733Global Solutions for a Nonlocal Problem with Logarithmic Source TermEugenio Lapa0Instituto de Investigacion ,FCM-UNMSM, Av. Venezuela S/N, Lima- Peru.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.https://scma.maragheh.ac.ir/article_712733_621d736ee4c56b9ae3237a0bb849f3d1.pdfglobal solutionsdegenerate nonlocal problemasymptotic behavior |
| spellingShingle | Eugenio Lapa Global Solutions for a Nonlocal Problem with Logarithmic Source Term Sahand Communications in Mathematical Analysis global solutions degenerate nonlocal problem asymptotic behavior |
| title | Global Solutions for a Nonlocal Problem with Logarithmic Source Term |
| title_full | Global Solutions for a Nonlocal Problem with Logarithmic Source Term |
| title_fullStr | Global Solutions for a Nonlocal Problem with Logarithmic Source Term |
| title_full_unstemmed | Global Solutions for a Nonlocal Problem with Logarithmic Source Term |
| title_short | Global Solutions for a Nonlocal Problem with Logarithmic Source Term |
| title_sort | global solutions for a nonlocal problem with logarithmic source term |
| topic | global solutions degenerate nonlocal problem asymptotic behavior |
| url | https://scma.maragheh.ac.ir/article_712733_621d736ee4c56b9ae3237a0bb849f3d1.pdf |
| work_keys_str_mv | AT eugeniolapa globalsolutionsforanonlocalproblemwithlogarithmicsourceterm |