A comparison principle for nonlinear parabolic equations with nonlocal source and gradient absorption

TL;DR

This paper establishes a comparison principle for a complex nonlinear parabolic equation with nonlocal and gradient terms, leading to new results on solution blow-up and boundedness, extending previous research in the field.

Contribution

It introduces a comparison principle for nonlinear parabolic equations with nonlocal sources and gradient absorption, unifying and extending existing theoretical results.

Findings

Established a comparison principle for the equation.

Derived conditions for solution blow-up.

Proved global boundedness of solutions under certain parameters.

Abstract

This paper investigates the initial-boundary value problem for a nonlinear parabolic equation involving the $p$-Laplacian operator, nonlocal source terms, gradient absorption, and various nonlinearities: \[ \frac{\partial u}{\partial t} - \text{div}(|\nabla u|^{p-2} \nabla u ) = \alpha |u|^{k-1}u \int_\Omega |u|^s \, dx - \beta |u|^{l-1}u |\nabla u|^q + \gamma u^m + \mu |\nabla u|^r - \nu |u|^{\sigma-1}u, \] where $ \Omega $ is a bounded domain in $\mathbb{R}^N$, $N \geq 1$, with a smooth boundary $\partial \Omega$. The parameters satisfy $ \alpha, l, \sigma > 0 $, $ \beta, \nu \geq 0 $, $ k, m, s \geq 1 $, $ r \geq p - 1 \geq \frac{p}{2}$, and $\gamma, \mu \in \mathbb{R}$. We establish a comparison principle for this problem. Using this principle, we derive blow-up results as well as global-in-time boundedness of solutions. Our results extend and unify previous studies in the literature.