Stability of solutions to abstract evolution equations with delay

An equation u̇=A(t)u+B(t)F(t,u(t−τ)), u(t)=v(t),−τ≤t≤0u(t)=v(t),−τ≤t≤0, is considered, where A(t)A(t) and B(t)B(t) are linear operators in a Hilbert space HH, u̇=dudt, F:H→HF:H→H is a non-linear operator, and τ>0τ>0 is a constant. Under some assumptions on A(t),B(t)A(t),B(t) and F(t,u)F(t,u) sufficient conditions are given for the solution u(t)u(t) to exist globally, i.e., for all t≥0t≥0, to be globally bounded, and to tend to zero at a specified rate as t→∞t→∞.