Quasistationary trajectories of semilinear dynamical equations of Sobolev type

The unique solvability of the Cauchy problem $u(0)=u_0$ for an operator differential equation $L\dot u=M(u)$ is investigated in the case of $L$-boundedness of the operator $M_{u_0}'$ . The results obtained are illustrated on the Cauchy–Dirichlet problem for the Hoff equation and for Oskolkov's system of equations.