The backward problem for time fractional evolution equations

In this paper, we consider the backward problem for fractional in time evolution equations ${\mathrm{\partial }}_t^{\alpha }u(t)=Au(t)$ 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 Hölder 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.