Nonlocal time porous medium equation with fractional time derivative

TL;DR

This paper studies nonlinear nonlocal diffusion equations with fractional derivatives and porous medium nonlinearities, proving existence, uniqueness, boundedness, and regularity of solutions using advanced mathematical techniques.

Contribution

It introduces a framework for analyzing fractional nonlocal porous medium equations, establishing key properties of solutions with novel mathematical methods.

Findings

Existence and uniqueness of weak solutions are proven.

Solutions are shown to be bounded and Hölder continuous.

The analysis employs maximal monotone operator theory and De Giorgi-Nash-Moser technique.

Abstract

We consider nonlinear nonlocal diffusive evolution equations, governed by fractional Laplace-type operators, fractional time derivative and involving porous medium type nonlinearities. Existence and uniqueness of weak solutions are established using approximating solutions and the theory of maximal monotone operators. Using the De Giorgi-Nash-Moser technique, we prove that the solutions are bounded and H\"{o}lder continuous for all positive time.