Opuscula Math. 43, no. 6 (2023), 741-758
https://doi.org/10.7494/OpMath.2023.43.6.741

 
Opuscula Mathematica

Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term

Aisha Alshehri
Noha Aljaber
Haya Altamimi
Rasha Alessa
Mohamed Majdoub

Abstract. 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.

Keywords: nonlinear heat equation, forcing term, blow-up, test-function, differential inequalities.

Mathematics Subject Classification: 35K05, 35A01, 35B44.

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  • Aisha Alshehri
  • Department of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi Arabia
  • Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 31441, Dammam, Saudi Arabia
  • Noha Aljaber
  • Department of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi Arabia
  • Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 31441, Dammam, Saudi Arabia
  • Haya Altamimi
  • Department of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi Arabia
  • Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 31441, Dammam, Saudi Arabia
  • Rasha Alessa
  • Department of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi Arabia
  • Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 31441, Dammam, Saudi Arabia
  • Mohamed Majdoub (corresponding author)
  • ORCID iD https://orcid.org/0000-0001-6038-1069
  • Department of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi Arabia
  • Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 31441, Dammam, Saudi Arabia
  • Communicated by Vicenţiu D. Rădulescu.
  • Received: 2023-05-03.
  • Revised: 2023-05-25.
  • Accepted: 2023-06-12.
  • Published online: 2023-07-22.
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Cite this article as:
Aisha Alshehri, Noha Aljaber, Haya Altamimi, Rasha Alessa, Mohamed Majdoub, Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term, Opuscula Math. 43, no. 6 (2023), 741-758, https://doi.org/10.7494/OpMath.2023.43.6.741

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