Well-posedness for weak and strong solutions of non-homogeneous initial boundary value problems for fractional diffusion equations

TL;DR

This paper establishes the well-posedness of both weak and strong solutions for non-homogeneous initial boundary value problems involving fractional diffusion equations, introducing new definitions and optimal conditions for solution existence.

Contribution

It introduces a novel definition of weak solutions and an optimal compatibility condition, expanding the understanding of solution existence for fractional diffusion equations.

Findings

Existence of weak solutions with non-zero initial and boundary data.

Existence of strong solutions under an optimal compatibility condition.

Conditions for higher regularity solutions in time and space.

Abstract

We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a new definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous terms lying in some arbitrary negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.