On nonlocal solutions of semilinear equations of the Sobolev type (Q354531)

The Cauchy problem \[ u(t_0) = u_0 \] is analyzed for the semilinear equation \[ L\dot{u}(t) = Mu(t) + N (t, u(t)), \] where \(L\) is a linear continuous operator, \(M\) is a closed linear operator and \(N\) is a nonlinear operator on the interval \(t \in (t_0, T)\). The existence and uniqueness of a nonlocal classical solution of this strongly degenerate equation, which is unsolvable for the derivative, are proved. The strong degeneracy of the equation is understood in the sense that the kernel of \(L\) contains a vector that has chains of \(M\)-associated vectors. The approach of the present paper simultaneously uses the theory of degenerate operator semigroups and results on the solvability of semilinear equations solved for the time derivative. The obtained abstract results are used to study initial-boundary value problems for some systems of equations unsolvable for the time derivative.