Stability result for abstract evolution problems

Consider an abstract evolution problem in a Hilbert space H u̇ = A(t)u+G(t, u) + f(t), u(0) = u0, (1) where A(t) is a linear, closed, densely defined operator in H with domain independent of t ≥ 0, G(t, u) is a nonlinear operator such that ||G(t, u)|| ≤ a(t)||u||, p = const > 1, ||f(t)|| ≤ b(t). We allow the spectrum of A(t) to be in the right half-plane Re(λ) < λ0(t), λ0(t) > 0, but assume that limt→∞λ0(t) = 0. Under suitable assumption on a(t) and b(t) we prove boundedness of ||u(t)|| as t→∞. If f(t) = 0, the Lyapunov stability of the zero solution to problem (1) with u0 = 0 is established. For f 6= 0, sufficient conditions for Lyapunov stability are given. The novel point in the paper is the possibility for the linear operator A(t) to have spectrum in the half-plane Re(λ) < λ0(t) with λ0(t) > 0 and limt→∞λ0(t) = 0 at a suitable rate.