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The representation by the Mittag-Leffler function of the solution to the Cauchy problem for the evolution equation solved for the higher derivative is used in the study of degenerate linear and quasilinear evolution equations under some special constraints on the nonlinear part of the equation. The solvability conditions for the Cauchy problem are simplified in the situation when the generalized Showalter-Sidorov condition is used as the initial condition. These results are applied to studying an initial boundary value problem for the motion equation of the Kelvin-Voigt fluid. [ABSTRACT FROM AUTHOR]