The paper deals with the problem of non-continuation for a solution of the abstract evolution equation

${\left[P\left({u}^{\text{'}}\left(t\right)\right)\right]}^{\text{'}}+A\left(u\left(t\right)\right)+Q(t,{u}^{\text{'}}\left(t\right))=F\left(u\left(t\right)\right)$,

$t\in (0,T),$ satisfying given initial data, where

$A,F,P$ and

$Q$ are nonlinear operators on appropriate Banach spaces. Under specific assumptions on these operators, the author proves, that the solution cannot exist for all time in the case of certain initial data. The results are applicable to important dynamical problems of nonlinear equations of elasticity with nonlinear damping.