Sobolev spaces and trace theorems for time-fractional evolution equations

TL;DR

This paper develops trace and extension theorems for time-fractional evolution equations in weighted Sobolev and Besov spaces, enhancing understanding of solution spaces and initial conditions for fractional diffusion models.

Contribution

It introduces new weighted Sobolev and Besov space frameworks tailored for time-fractional evolution equations, including sub-diffusion and super-diffusion types.

Findings

Established trace and extension theorems in weighted spaces

Identified solution spaces suitable for fractional derivatives

Provided insights into initial behavior of solutions

Abstract

We establish trace and extension theorems for evolutionary equations with the Caputo fractional derivatives in (weighted) $L_p$ spaces. To achieve this, we identify weighted Sobolev and Besov spaces with mixed norms that accommodate solution spaces and their initial values well-suited for equations involving time-fractional derivatives. Our analysis encompasses both time-fractional sub-diffusion and super-diffusion equations. We also provide observations on the initial behavior of solutions to time-fractional equations.