# Time fractional diffusion equations: solution concepts, regularity and long-time behaviour

## TL;DR

This paper surveys analytical results on time fractional diffusion equations, focusing on solution concepts, regularity, and long-time behavior, highlighting differences from classical heat equations.

## Contribution

It compiles recent advances on solution theories, including strong and weak solutions, and discusses the long-time dynamics of fractional diffusion models.

## Key findings

- Strong solutions in $L_p$ sense established
- Weak solutions for rough coefficients analyzed
- Distinct long-time behavior compared to heat equations

## Abstract

In this paper we give a survey of results on various analytical aspects of time fractional diffusion equations. We describe the approach via abstract Volterra equations and collect results on strong solutions in the $L_p$ sense. We further discuss the concept of weak solutions for equations with rough coefficients and give an account of recent developments towards a De Giorgi-Nash-Moser theory for such equations. The last part summarizes recent results on the long-time behaviour of solutions, which turns out to be significantly different from that in the heat equation case.