An identification problem is considered for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Solutions of problems with Cauchy and Showalter conditions on initial values are proved to be existing and unique. Solutions stability estimates are derived. The abstract results are applied to an identification problem for the linearized Oskolkov system of equations. There are considered different degrees of system degeneration with respect to the time derivatives of unknown functions.

A nonlinear inverse problem for an equation in a Banach space, resolved with respect to the oldest Gerasimov–Caputo fractional derivative, is considered. A linear closed operator in the linear part of the equation generates an analytic resolving family of operators. Nonlinear operator in the equation depends on minor fractional derivatives, fractional integrals and an unknown element, which depends on time. The equation is endowed by the Cauchy conditions and an overdetermination condition with an abstract linear bounded operator. By the method of a contraction mapping in a metric space theorems on the unique solvability in the sense of generalized solution and of smooth solution are proved. Abstract results were used for research of a nonlinear inverse problem for a system of Navier–Stokes type with time-fractional Gerasimov–Caputo derivatives and integrals.

The purpose of this work is to obtain sufficient conditions of a solution existence and uniqueness for a class of inverse problems for linear evolution equations with a degenerate operator at the derivative and with an unknown element in the right-hand side of the equation, which depends on the time variable. The overdetermination condition is given on the kernel of the operator at the derivative, the initial condition have the Cauchy form or the Showalter–Sidorov form. The obtained abstract results are applied to the investigation of linear inverse problems for the Sobolev system of equations and for the linearized Oskolkov system with overdetermination on the pressure gradient function.