Source inverse problem for fractional kinetic equation

We study the Cauchy problem for fractional kinetic equation in spaces with values in the entire scale of spaces of Bessel potentials, prove its unique, classical in time, solvability, obtaining an analogue of the maximum regularity of the solution according to Da Prato and Grisvard.We also study the inverse problem on determining a space-dependent component in the right-hand side of such equation under a time-integral additional condition. We use the method of fundamental solution of the equation.We are investigating properties of the Green's operators of the Cauchy problem for fractional kinetic equation in spaces of continuous functions with values in spaces of Bessel potentials and find sufficient conditions for a time-local, classical in time and with values in spaces of Bessel potentials, unique solvability of the inverse problem. We define the unknown function on the right-hand side of the fractional kinetic equation over the entire scale of spaces of Bessel potentials. The solution of the problem is reduced to the study of the unique solvability of some Fredholm's linear integral equation of the second kind in spaces of continuous functions with values in the entire scale of spaces of Bessel potentials. The unknown function on the right-hand side of the fractional kinetic equation is expressed through the solution of this integral equation.