Asymptotic Behavior

In Part II, we saw how to associate to a delay differential equation,

(DEp)

⎧⎪⎨⎪⎩ u′(t) = Bu(t) + Φut, t ≥ 0, u(0) = x, u0 = f,

an operator matrix

A := ( B Φ 0 ddσ

) D(A) :=

) ∈ D(B)×W 1,p([−1, 0], Z) : f(0) = x} , on the Banach space Ep = X × Lp([−1, 0], Z), 1 ≤ p < ∞, and how to discuss well-posedness of (DEp).