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

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: $$\frac{\mathrm{\partial }u}{\mathrm{\partial }t}-\text{div}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)=\alpha \mathrm{\mid }u{\mathrm{\mid }}^{k-1}u{\int }_{\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^{s}dx-\beta \mathrm{\mid }u{\mathrm{\mid }}^{l-1}u\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q+\gamma u^m+\mu \mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^r-\nu \mathrm{\mid }u{\mathrm{\mid }}^{\sigma -1}u,$$ where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$, $N\ge 1$, with a smooth boundary $\mathrm{\partial }\mathrm{\Omega }$. 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.