Stability of solutions to abstract evolution equations with delay

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 ̇ = d u d t , 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 → ∞ .