A stability result for abstract evolutionproblems

Consider an abstract evolution problem in a Hilbert space H u.=A(t)u+G(t,u)+f(t),u(0)=u0, where A(t) is a linear, closed, densely defined operator in H with domain independent of t0 and G(t,u) is a nonlinear operator such that G(t,u)a(t) up, p=const>1, f(t)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 limt0(t)=0. Under suitable assumptions on a(t) and b(t), the boundedness of u(t) as t is proved. If f(t)=0, the Lyapunov stability of the zero solution to problem (1) with u0=0 is established. For f0, 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 limt0(t)=0 at a suitable rate. The new technique, proposed in the paper, is based on an application of a novel nonlinear differential inequality. Copyright (c) 2012 John Wiley & Sons, Ltd.