A stability result for abstract evolution problems

where A.t/ is a linear, closed, densely defined operator in H with domain independent of t 0 and G.t, u/ is a nonlinear operator such that jjG.t, u/jj a.t/jjujjp, p D const > 1, jjf .t/jj b.t/. We allow the spectrum of A.t/ to be in the right half-plane Re. / < 0.t/, 0.t/ > 0, but assume that limt!1 0.t/D 0. Under suitable assumptions on a.t/ and b.t/, the boundedness of jju.t/jj as t ! 1 is proved. If f .t/ D 0, the Lyapunov stability of the zero solution to problem (1) with u0 D 0 is established. For f ¤ 0, sufficient conditions for the Lyapunov stability are given. The novel point in our study of the stability of the solutions to abstract evolution equations is the possibility for the linear operator A.t/ to have spectrum in the half-plane Re. / < 0.t/ with 0.t/ > 0 and limt!1 0.t/ D 0 at a suitable rate. The new technique, proposed in the paper, is based on an application of a novel nonlinear differential inequality. Copyright © 2012 John Wiley & Sons, Ltd.