An inverse problem of recovering the variable order of the derivative in a fractional diffusion equation

We consider a fractional diffusion equation with variable space-dependent order of the derivative in a bounded multidimensional domain. The initial data are homogeneous and the right-hand side and its time derivative satisfy some monotonicity conditions. Addressing the inverse problem with final overdetermination, we establish the uniqueness of a solution as well as some necessary and sufficient solvability conditions in terms of a certain constructive operator $A$. Moreover, we give a simple sufficient solvability condition for the inverse problem. The arguments rely on the Birkhoff–Tarski theorem.