TY  - JOUR
ID  - Alshehri2023
LB  - Alshehri2023
AU  - Alshehri, Aisha
AU  - Aljaber, Noha
AU  - Altamimi, Haya
AU  - Alessa, Rasha
AU  - Majdoub, Mohamed
TI  - Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term
ST  - Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2023
VL  - 43
IS  - 6
SP  - 741
EP  - 758
DO  - https://doi.org/10.7494/OpMath.2023.43.6.741
UR  - https://doi.org/10.7494/OpMath.2023.43.6.741
AB  - 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.
KW  - nonlinear heat equation
KW  - forcing term
KW  - blow-up
KW  - test-function
KW  - differential inequalities
SN  - 1232-9274
ER  -