An Inverse Source Problem in a Variable-Order Time-Fractional Diffusion PDE

We study an inverse source problem for a semilinear diffusion equation involving a Caputo-type time-fractional derivative whose order is a function of time. The equation is considered in a bounded Lipschitz domain Ω⊂Rd, d≥1, and is supplemented with homogeneous Dirichlet boundary conditions. The source term is taken to be separable, h(t)f(x), where the temporal component h(t) is unknown. This quantity is to be identified from spatially localized measurements m(t) of the solution. In this setting, we establish existence and uniqueness results in suitable function spaces, thereby demonstrating the well-posedness of the corresponding inverse source problem.