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 $\Omega$ is a bounded domain in ${\mathbb{R}}^N$, $N\ge 1$, with a smooth boundary $\partial \Omega$. The parameters satisfy $\alpha ,l,\sigma >0$, $\beta ,\nu \ge 0$, $k,m,s\ge 1$, $r\ge p-1\ge \frac{p}{2}$, 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.

14 pages