Nonlinear inverse problem for the Oskolkov system, linearized in a stationary solution neighborhood

A class of nonlinear inverse problems with unknown parameters depending on the time is researched for the abstract Sobolev type equation in Banach spaces. Along with the Cauchy condition, the generalized Showalter condition is used in the statement of the problem. Theorems of existence and uniqueness of mild and smooth solutions are proved. General results are applied to a nonlinear evolution problem for the linearized Oskolkov system modeling the dynamics of a viscoelastic fluid.