On the Fujita exponent for a nonlinear parabolic equation with a forcing term

TL;DR

This paper investigates the conditions under which solutions to a nonlinear parabolic equation blow up, focusing on the influence of a forcing term and extending previous results by analyzing a broader class of functions for this term.

Contribution

It extends earlier blow-up results for nonlinear parabolic equations by considering a wider class of forcing functions ${\mathtt a}(t)$, providing a more comprehensive understanding of the Fujita exponent.

Findings

Identified conditions for solution blow-up with various forcing terms.

Extended the range of functions ${\mathtt a}(t)$ for which blow-up occurs.

Improved understanding of the interplay between nonlinear terms and forcing in parabolic equations.

Abstract

The purpose of this work is to analyze the blow-up of solutions of the nonlinear parabolic equation \[ u_t-\Delta u=|x|^{\alpha}|u|^{p}+{\mathtt a}(t)\textbf{w}(x) \ \quad\mbox{for } (t,x)\in(0,\infty)\times\mathbb{R}^{N}, \] where $p>1$, $\alpha\in\mathbb{R}$ and ${\mathtt a}$, $\textbf{w}$ are suitable given functions. We improve earlier results by considering a wide class of functions ${\mathtt a}(t)$.