Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term
The purpose of this work is to analyze the blow-up of solutions of a nonlinear parabolic equation with a forcing term depending on both time and space variables \[u_t-\Delta u=|x|^{\alpha} |u|^{p}+{\mathtt a}(t)\,{\mathbf w}(x)\quad\text{for }(t,x)\in(0,\infty)\times \mathbb{R}^{N},\] where \(\alpha...
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| Format: | Article |
| Language: | English |
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AGH Univeristy of Science and Technology Press
2023-07-01
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| Series: | Opuscula Mathematica |
| Subjects: | |
| Online Access: | https://www.opuscula.agh.edu.pl/vol43/6/art/opuscula_math_4335.pdf |
| _version_ | 1797774666479173632 |
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| author | Aisha Alshehri Noha Aljaber Haya Altamimi Rasha Alessa Mohamed Majdoub |
| author_facet | Aisha Alshehri Noha Aljaber Haya Altamimi Rasha Alessa Mohamed Majdoub |
| author_sort | Aisha Alshehri |
| collection | DOAJ |
| description | The purpose of this work is to analyze the blow-up of solutions of a nonlinear parabolic equation with a forcing term depending on both time and space variables \[u_t-\Delta u=|x|^{\alpha} |u|^{p}+{\mathtt a}(t)\,{\mathbf w}(x)\quad\text{for }(t,x)\in(0,\infty)\times \mathbb{R}^{N},\] where \(\alpha\in\mathbb{R}\), \(p\gt 1\), and \({\mathtt a}(t)\) as well as \({\mathbf w}(x)\) are suitable given functions. We generalize and somehow improve earlier existing works by considering a wide class of forcing terms that includes the most common investigated example \(t^\sigma\,{\mathbf w}(x)\) as a particular case. Using the test function method and some differential inequalities, we obtain sufficient criteria for the nonexistence of global weak solutions. This criterion mainly depends on the value of the limit \(\lim_{t\to\infty} \frac{1}{t}\,\int_0^t\,{\mathtt a}(s)\,ds\). The main novelty lies in our treatment of the nonstandard condition on the forcing term. |
| first_indexed | 2024-03-12T22:24:25Z |
| format | Article |
| id | doaj.art-2a82c85f557f48128dd1f222e19b49e6 |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-03-12T22:24:25Z |
| publishDate | 2023-07-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-2a82c85f557f48128dd1f222e19b49e62023-07-22T07:15:49ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742023-07-01436741758https://doi.org/10.7494/OpMath.2023.43.6.7414335Nonexistence of global solutions for a nonlinear parabolic equation with a forcing termAisha Alshehri0Noha Aljaber1Haya Altamimi2Rasha Alessa3Mohamed Majdoub4https://orcid.org/0000-0001-6038-1069Department of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi ArabiaDepartment of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi ArabiaDepartment of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi ArabiaDepartment of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi ArabiaDepartment of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi ArabiaThe purpose of this work is to analyze the blow-up of solutions of a nonlinear parabolic equation with a forcing term depending on both time and space variables \[u_t-\Delta u=|x|^{\alpha} |u|^{p}+{\mathtt a}(t)\,{\mathbf w}(x)\quad\text{for }(t,x)\in(0,\infty)\times \mathbb{R}^{N},\] where \(\alpha\in\mathbb{R}\), \(p\gt 1\), and \({\mathtt a}(t)\) as well as \({\mathbf w}(x)\) are suitable given functions. We generalize and somehow improve earlier existing works by considering a wide class of forcing terms that includes the most common investigated example \(t^\sigma\,{\mathbf w}(x)\) as a particular case. Using the test function method and some differential inequalities, we obtain sufficient criteria for the nonexistence of global weak solutions. This criterion mainly depends on the value of the limit \(\lim_{t\to\infty} \frac{1}{t}\,\int_0^t\,{\mathtt a}(s)\,ds\). The main novelty lies in our treatment of the nonstandard condition on the forcing term.https://www.opuscula.agh.edu.pl/vol43/6/art/opuscula_math_4335.pdfnonlinear heat equationforcing termblow-uptest-functiondifferential inequalities |
| spellingShingle | Aisha Alshehri Noha Aljaber Haya Altamimi Rasha Alessa Mohamed Majdoub Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term Opuscula Mathematica nonlinear heat equation forcing term blow-up test-function differential inequalities |
| title | Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term |
| title_full | Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term |
| title_fullStr | Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term |
| title_full_unstemmed | Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term |
| title_short | Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term |
| title_sort | nonexistence of global solutions for a nonlinear parabolic equation with a forcing term |
| topic | nonlinear heat equation forcing term blow-up test-function differential inequalities |
| url | https://www.opuscula.agh.edu.pl/vol43/6/art/opuscula_math_4335.pdf |
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