Linear Functional Differential Equations as Semigroups on Product Spaces

In this paper we consider the well-posedness of linear functional differential equations on product spaces. Let L and D be linear Rⁿ-valued functions with domains D(L) and D(D) subspaces of the Lebesgue measurable Rⁿ-valued functions on $[ - r,0]$ and such that W1,p ([ - r,0];Rⁿ ) ⊆ D(L) ∩ D(D). Under weak conditions on D and L we establish the equivalence between generalized solutions to the functional differential equation \[ {d}{{dt}}Dx_t = Lx_t + f(t)\] and mild solutions to the Cauchy problem in Rⁿ × Lₚ ([ - r,0];Rⁿ )\[ z(t) = a z(t) + (f(t),0), \] where a is the operator defined on \[ D(a) = \{ {(η ,φ ) ∈ R^n × L_p {{( {[ - r,0];R^n } )} / φ } ∈ W1,p ( {[ - r,0];R^n } ),Dφ = η } \},\] by \[ a(η ,φ ) = (Lφ , φ ). \] The results are applicable to neutral functional differential equations and certain singular integral equations.