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

TL;DR

This work analyzes an initial-boundary value problem for a nonlinear parabolic equation with a $p$-Laplacian, nonlocal source, gradient absorption, and multiple nonlinear terms. It establishes a robust comparison principle for weak solutions and uses it to derive both finite-time blow-up and global-in-time boundedness results, employing explicit sub- and super-solution constructions and careful term-wise estimates. By treating all nonlinear effects simultaneously, the results unify and extend prior analyses and identify parameter regimes governing extreme behaviors. The methods illuminate the diffusion–reaction–absorption interplay in nonlocal nonlinear PDEs and provide tools for predicting long-term dynamics in related models.

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.

Theorems & Definitions (8)

• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• proof : Proof of Theorem \ref{['thm_comp_princ']}
• Theorem 3.1
• proof : Proof of Theorem \ref{['blow_thm']}
• Theorem 3.2
• proof : Proof of Theorem \ref{['thm_glob_exis_1']}