5. Inverse Problems of Fractional Wave Equations

In this chapter, we firstly concern with a backward problem (or called initial inverse problem) for a nonlinear time fractional wave equation in a bounded domain. By applying the properties of Mittag–Leffler functions and the method of eigenfunction expansion, we establish some results about the existence and uniqueness of 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. In Sect. 5.2, we consider the backward problem for an inhomogeneous time fractional wave equation in a general bounded domain. We show that the backward problem is ill-posed, and we propose a regularizing scheme by using a fractional Landweber regularization method. We also present error estimates between the regularized solution and the exact solution under two parameter choice rules. In Sect. 5.3, we consider the terminal value problem of determining the initial value, in a general class of time fractional wave equation with the Caputo derivative, from a given final value. We are concerned with the existence and regularity upon the terminal value data of the mild solution. Under some assumptions of the nonlinear source function, we address and show the well-posedness for the terminal value problem. Some regularity results for the mild solution and its derivatives of first and fractional orders are also derived. The effectiveness of our methods is shown by applying the results to two interesting models: time fractional Ginzburg–Landau equation and time fractional Burgers equation, where time and spatial regularity estimates are obtained. The contents of Sect. 5.1 are taken from He and Zhou (Proc R. Soc Edinburgh Sect A 152(6):1589–1612, 2022). The results in Sect. 5.2 are adopted from Huynh et al. (Appl Anal 100(4):860–878, 2021). Section 5.3 is from Tuan et al. (Nonlinearity 34(3):1448, 2021).