Resolving operators of a linear degenerate evolution equation with Caputo derivative. The sectorial case

Unique solvability of the Cauchy problem for an equation in a Banach space with degenerate operator at the fractional Caputo derivative is studied. Previously found conditions of the existence of analytic in a sector resolving operators family for the equation with the degeneration on the kernel of the operator at the derivative is used. The form of the resolving operators is established. Under the satisfied conditions, the existence is shown for the unique solution of the Cauchy problem to the researched equation with initial data from the complement of the kernel of the operator at the derivative and the solution is presented using the resolving operators. The obtained results are applied to studying the linearized quasistationary time-fractional order system of the phase field equations.