Well-posedness of linear partial differential equations with unbounded delay operators ✩

In this paper we show the well-posedness of the following constant delay equation: ˙u(t ) = Au(t )+ Bu(t − r), t >0, u(0) = x, u(θ) = f (θ), θ ∈ [−r, 0], where (A,D(A)) generates a strongly continuous semigroup (T (t ))t 0 on a Banach space X and (B,D(B)) is an unbounded and closed linear operator on X. Our approach is to transform the delay equation into an abstract Cauchy problem in a special phase space and apply semigroup theory. Furthermore, we show that the solution semigroup corresponding to the above delay equation is uniformly exponentially bounded. Finally, we give an example to explain the well-posedness of delay equations.  2004 Elsevier Inc. All rights reserved