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

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.