Qualitative properties of solutions to a nonlinear time-space fractional diffusion equation

TL;DR

This paper investigates the qualitative behavior of solutions to a nonlocal nonlinear diffusion equation involving fractional derivatives, establishing comparison principles, existence, blow-up, and asymptotic properties.

Contribution

It provides a simple algebraic proof of the comparison principle and classifies solution behaviors for a fractional nonlinear diffusion equation.

Findings

Comparison principle proved using algebraic relations.

Existence of local weak solutions established via Galerkin method.

Classification of blow-up and global solutions, including asymptotic behavior.

Abstract

In the present paper, we study the Cauchy-Dirichlet problem to the nonlocal nonlinear diffusion equation with polynomial nonlinearities $$\mathcal{D}_{0|t}^{\alpha }u+(-\Delta)^s_pu=\gamma|u|^{m-1}u+\mu|u|^{q-2}u,\,\gamma,\mu\in\mathbb{R},\,m>0,q>1,$$ involving time-fractional Caputo derivative $\mathcal{D}_{0|t}^{\alpha}$ and space-fractional $p$-Laplacian operator $(-\Delta)^s_p$. We give a simple proof of the comparison principle for the considered problem using purely algebraic relations, for different sets of $\gamma,\mu,m$ and $q$. The Galerkin approximation method is used to prove the existence of a local weak solution. The blow-up phenomena, existence of global weak solutions and asymptotic behavior of global solutions are classified using the comparison principle.