Comparison principles for the time-fractional diffusion equations with the Robin boundary conditions. Part I: Linear equations

TL;DR

This paper analyzes initial-boundary value problems for linear time-fractional diffusion equations with Robin boundary conditions, establishing existence, uniqueness, positivity, and comparison principles in fractional Sobolev spaces.

Contribution

It provides new theoretical results on the well-posedness and comparison principles for linear time-fractional diffusion equations with Robin boundary conditions.

Findings

Proved existence and uniqueness of solutions.

Established positivity of solutions.

Derived comparison principles for solutions.

Abstract

The main objective of this paper is analysis of the initial-boundary value problems for the linear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type time-fractional derivative acting in the fractional Sobolev spaces. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. First we discuss the uniqueness and existence of solutions to these initial-boundary value problems. Under some suitable conditions on the problem data, we then prove positivity of the solutions. Based on these results, several comparison principles for the solutions to the initial-boundary value problems for the linear time-fractional diffusion equations are derived.