Local and Global Solvability of Nonlinear Inverse Problems for Degenerate Evolution Equations with Fractional Derivatives

Nonlinear inverse problems with a time-dependent parameter are considered both for equations solvable with respect to the Gerasimov–Caputo leading fractional derivative and for equations with a degenerate operator at the derivative. The global existence of a unique generalized solution to the nonlinear inverse problem is proven for a nondegenerate equation with a sectorial operator in the linear part. In the degenerate case, under the condition that the image of the nonlinear operator belongs to a subspace without degeneration, the local existence and uniqueness of the generalized and smooth solutions and the global existence and uniqueness of the generalized solution to the nonlinear inverse problem are obtained. Under the condition that the nonlinear operator is independent of the elements of the degeneration subspace, the local and global existence of a unique generalized solution to the inverse problem are also proven. The results thus obtained are illustrated by examples of coefficient inverse problems for degenerate systems of partial differential equations.