,Vonlinear Anolvsrs, Theorv. Mrrhods & Applrcatrons, Vol 4. No 2. PP. 41 I-414 0 Pergamon Press Ltd. 1980. Printed in Great Br~tarn 0362.54h\ 80~02ul-011 I 102 ooio zyxwvutsr EXISTENCE AND REGULARITY FOR LINEAR DELAY PARTIAL DIFFERENTIAL EQUATIONS A. ARDITO* and P. RICCIARDI* Istituto Matematico “G. Castelnuovo” Universitl di Roma Italy zyxwvutsrqponmlkjihgfedcb (Received 18 April 1979) Key wor&: Analytic semigroup, delay partial differential equation, interpolation spaces. 1. INTRODUCTION IN THIS PAPER we study the problem (P) Here w > 0 and h: [0, w] + X are given. A is the infinitesimal group {etA, t > 0}, and B is a regular enough linear operator. In problem f u(t) = e(g-“‘Ah(w) + s e” - s’AB~(~ _ w generator of the analytic semi- addition, we consider the ‘mild’ w) ds and, if A-‘B is preclosed, set T = A-‘B its closure, we consider s f 1 u(t) = et’- “jAh(w) + A e(t-s)A Tu(s - w) ds w (P,) u(t) = h(t), fE[O,W]. We observe that any solution of problem (P) is a solution of (P,) and that if B is continuous, any solution of(P) is a solution of (P,). We study the problems (P,) and (P,) using some interpolation spaces introduced by Da Prato and Grisvard [2]. We are able to use the classical method of steps without losing in regularity at each step. In the particular case where B is dominated by A our results for the problem (P,) are similar to that of Travis and Webb [I]. The results of this paper can be extended to the nonlinear case as we will show in a future paper. 2. PRELIMINARIES Let X be a Banach space and let 1). (/denote its norm. Let (H,) {A: DA t X + X be the infinitesimal generator of the analytic semigroup erA,t > 0 in X. DA is a Banach space with the graph norm. __ * Work supported by GNAFA-CNR. 411