The backward problem for time fractional evolution equations

TL;DR

This paper investigates the backward problem for fractional time evolution equations, establishing stability estimates and extending techniques to fractional derivatives with applications to various problems.

Contribution

It extends the logarithmic convexity method to fractional equations and derives new stability estimates for the backward problem.

Findings

Established conditional stability estimates of Hölder type.

Extended logarithmic convexity technique to fractional derivatives.

Applied results to multiple fractional evolution problems.

Abstract

In this paper, we consider the backward problem for fractional in time evolution equations $\partial_t^\alpha u(t)= A u(t)$ with the Caputo derivative of order $0<\alpha \le 1$, where $A$ is a self-adjoint and bounded above operator on a Hilbert space $H$. First, we extend the logarithmic convexity technique to the fractional framework by analyzing the properties of the Mittag-Leffler functions. Then we prove conditional stability estimates of H\"older type for initial conditions under a weaker norm of the final data. Finally, we give several applications to show the applicability of our abstract results.