# Variational formulation of time-fractional parabolic equations

## TL;DR

This paper develops a variational framework for time-fractional parabolic PDEs of order between 0 and 1, establishing well-posedness using fractional Sobolev-Bochner spaces and addressing initial value considerations.

## Contribution

It introduces a novel variational formulation for fractional parabolic equations based on fractional Sobolev-Bochner spaces, clarifying initial value issues.

## Key findings

- Proves well-posedness of the variational formulation.
- Clarifies the role of initial values in fractional PDEs.
- Establishes a rigorous mathematical foundation for fractional diffusion equations.

## Abstract

We consider initial/boundary value problems for time-fractional parabolic PDE of order $0<\alpha<1$ with Caputo fractional derivative (also called fractional diffusion equations in the literature). We prove well-posedness of corresponding variational formulations based entirely on fractional Sobolev-Bochner spaces, and clarify the question of possible choices of the initial value.