Semilinear degenerate evolution equations and nonlinear systems of hydrodynamic type

Methods from the theory of degenerate semigroups of operators are used to prove the local existence and uniqueness of solutions to the Cauchy and Showalter problems for some new classes of semilinear first-order differential equations in a Banach space with degenerate operator at the derivative and nonstationary nonlinear operator at the required function. The obtained general results are used in the investigation of solvability of initial-boundary value problems for a class of systems of equations of generalized hydrodynamic type including Oskolkov's system of equations for the dynamics of viscoelastic fluid and its complicated versions, for example, with nonstationary nonlinearity, with nonlinear viscosity, weighted systems, etc.