On existence and regularity of a terminal value problem for the time fractional diffusion equation

TL;DR

This paper investigates the existence and regularity of solutions to a terminal value problem for a time-space fractional diffusion equation, utilizing spectral methods and Laplace transforms.

Contribution

It provides new existence and regularity results for fractional diffusion equations with both linear and nonlinear cases, expanding understanding of such problems.

Findings

Representation of solutions via Laplace transform and spectrum.

Existence of solutions under certain conditions.

Regularity results for solutions in fractional diffusion context.

Abstract

In this paper we consider a final value problem for a diffusion equation with time-space fractional differentiation on a bounded domain $D$ of $ \mathbb{R}^{k}$, $k\ge 1$, which includes the fractional power $\mathcal L^\beta$, $0<\beta\le 1$, of a symmetric uniformly elliptic operator $\mathcal L$ defined on $L^2(D)$. A representation of solutions is given by using the Laplace transform and the spectrum of $\mathcal L^\beta$. We establish some existence and regularity results for our problem in both the linear and nonlinear case.