An equation u=A(t)u+B(t)F(t,u(t-τ)), u(t)=v(t),-τ≤t≤0, is considered, where A(t) and B(t) are linear operators in a Hilbert space H, u=dudt, F: H→ H is a non-linear operator, and τ > 0 is a constant. Under some assumptions on A(t). , B(t) and F(t, u) sufficient conditions are given for the solution u(t) to exist globally, i.e., for all t≥ 0, to be globally bounded, and to tend to zero at a specified rate as t→ ∞. © 2012 Elsevier Ltd.
Ramm, A. G. (2012). Stability of solutions to abstract evolution equations with delay. Journal of Mathematical Analysis and Applications, 396(2), 523–527. https://doi.org/10.1016/j.jmaa.2012.06.033