On a backward problem for nonlinear time fractional wave equations

In this paper, we concern with a backward problem for a nonlinear time fractional wave equation in a bounded domain. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we establish some results about the existence and uniqueness of the mild solutions of the proposed problem based on the compact technique. Due to the ill-posedness of backward problem in the sense of Hadamard, a general filter regularization method is utilized to approximate the solution and further we prove the convergence rate for the regularized solutions.

Weak Solutions for Time-Fractional Evolution Equations in Hilbert Spaces

Our purpose is to introduce a notion of weak solution for a class of abstract fractional differential equations. We point out that the time fractional derivative occurring in the equations is in the sense of the Caputo derivative. We prove existence results for weak and strong solutions. To justify the abstract theory we develop, we apply two examples of concrete equations: time-fractional wave equations and time-fractional Petrovsky systems. Both these concrete examples are of great interest in the theory of fractional partial differential equations.