Determination of a coefficient and kernel in a $d$-dimensional fractional integro-differential equation

This paper is devoted to obtaining a unique solution to an inverse problem for a multidimensional time-fractional integro-differential equation. In the case of additional data, we consider an inverse problem. The unknown coefficient and kernel are uniquely determined by the additional data. By using the fixed point theorem in suitable Sobolev spaces, the global in time existence and uniqueness results of this inverse problem are obtained. The weak solvability of a nonlinear inverse boundary value problem for a $d$-dimensional fractional diffusion-wave equation with natural initial conditions was studied in the work. First, the existence and uniqueness of the direct problem were investigated. The considered problem was reduced to an auxiliary inverse boundary value problem in a certain sense and its equivalence to the original problem was shown. Then, the local existence and uniqueness theorem for the auxiliary problem is proved using the Fourier method and contraction mappings principle. Further, based on the equivalency of these problems, the global existence and uniqueness theorem for the weak solution of the original inverse coefficient problem was established for any value of time.