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

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: \[ \frac{\partial u}{\partial t} - \text{div}(|\nabla u|^{p-2} \nabla u ) = α|u|^{k-1}u \int_Ω|u|^s \, dx - β|u|^{l-1}u |\nabla u|^q + γu^m + μ|\nabla u|^r - ν|u|^{σ-1}u, \] where $ Ω$ is a bounded domain in $\mathbb{R}^N$, $N \geq 1$, with a smooth boundary $\partial Ω$. The parameters satisfy $ α, l, σ> 0 $, $ β, ν\geq 0 $, $ k, m, s \geq 1 $, $ r \geq p - 1 \geq \frac{p}{2}$, and $γ, μ\in \mathbb{R}$. 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.