3. Inverse Problems of Fractional Diffusion Equations

This chapter deals with the inverse problems of time fractional diffusion equations of order \(\alpha \in (0,1)\). In Sect. 3.1, we study a backward problem for an inhomogeneous fractional diffusion equation in a bounded domain. By applying the properties of the Mittag-Leffler functions and the method of eigenvalue expansion, we establish some results about the existence, uniqueness, and regularity of the mild solutions and the classical solutions of the proposed problem in a weighted Hölder continuous function space. In Sect. 3.2, we consider a final value problem for a diffusion equation with time-space fractional differentiation on a bounded domain D of \( \mathbb {R}^{k}\), \(k\ge 1\), which includes the fractional power \(\mathscr {L}^\beta \), \(0<\beta \leq 1\), of a symmetric uniformly elliptic operator \(\mathscr {L}\) defined on \(L^2(D)\). A representation of solutions is given by using the Laplace transform and the spectrum of \(\mathscr {L}^\beta \). We present some existence and regularity results for our problem in both the linear and nonlinear cases. The materials in Sect. 3.1 are adopted from Zhou, He, Ahmad, and Tuan [245]. The contents in Sect. 3.2 are due to Tuan, Ngoc, Zhou, and O’Regan [210].