Diffusion equations with general nonlocal time and space derivatives

TL;DR

This paper derives and analyzes general diffusion equations with nonlocal time and space derivatives based on CTRW theory, proving existence, positivity, and boundedness of solutions, and extending results to bounded domains.

Contribution

It introduces a generalized diffusion equation with nonlocal derivatives and proves solution existence and properties, extending previous fractional Laplacian frameworks.

Findings

Existence of solutions for the Cauchy problem is established.

Solutions are shown to be positive and bounded.

Existence results are extended to bounded domains using Friedrichs extension.

Abstract

In the present study, firstly, based on the continuous time random walk (CTRW) theory, general diffusion equations are derived. The time derivative is taken as the general Caputo-type derivative introduced by Kochubei and the spatial derivative is the general Laplacian defined by removing the conditions (1.5) and (1.6) from the definition of the general fractional Laplacian proposed in the paper (Servadei and Valdinoci, 2012). Secondly, the existence of solutions of the Cauchy problem for the general diffusion equation is proved by extending the domain of the general Laplacian to a general Sobolev space. The results for positivity and boundedness of the solutions are also obtained. In the last, the existence result for solutions of the initial boundary value problem (IBVP) for the general diffusion equation on a bounded domain is established by using the Friedrichs extension of the general fractional Laplacian introduced in the book (Bisci, Radulescu and Servadei, 2016).