Inverse initial-value problems for time fractional diffusion equations in fractional Sobolev spaces

We study the time fractional diffusion equation ${\partial }_{t}u={\partial }_t^{1-\alpha }Au+G\left(u\right)$, $0<\alpha <1$, in a bounded domain $\Omega \subset R^N$ with an elliptic operator $A$ and a locally Lipschitz nonlinearity $G$ on fractional Sobolev spaces, subjected to the homogeneous Dirichlet boundary condition. Data have not been measured at the initial time $t=0$, but at a final time $T>0$, that is, $u\left(T\right)$ is given instead of $u\left(0\right)$. The problem is, therefore, called an inverse initial-value problem. We first establish the well-posedness of this problem on fractional Sobolev spaces and the regularity of the solution by assuming only the local Lipschitz continuity of $G$. Second, an susceptible-infected (shortly, SI) model with heterogeneity and a Navier–Stokes equation have been exemplified. Finally, a spatial $L^{\infty }$-estimate for the solution and its gradient has been provided. The essential tools are asymptotic behaviours of Mittag–Leffler functions, fractional power spaces, fractional Sobolev spaces and embedding, weighted functional spaces, and $L^r-L^s$ estimates for heat semigroup.