Backward problem for time-space fractional diffusion equations in Hilbert scales

This work is concerned with a mathematical study of backward problem for time-space fractional diffusion equations associated with the observed data measured in Hilbert scales. Transforming the original problem into an operator equation, we investigate the existence, the uniqueness and the instability for the problem. In order to overcome the ill-posedness of the problem, we apply a modified version of quasi-boundary value method to construct stable approximation problem. Using a Hölder-type smoothness assumption of the exact solution it is shown that estimates achieve optimal rates of convergence in Hilbert scales both for an a-priori and for an a-posteriori parameter choice strategies.