Abstract evolution equations

Abstract

JEROME A. GOLDSTEINS) 1. Introduction.Let 3£ be a Banach space, and let sé be the collection of all infinitesimal generators of strongly continuous contraction semigroups of bounded linear operators on £.For A es/, 2¿(A) denotes the domain of A and {T(t; A), r^O} denotes the semigroup generated by A. Cauchy problems for abstract evolution equations of the form (1.1) du/dt = A(t)u, (t^s), u(s)=f are considered in this paper.In case A(t) ( = A) is independent of t, then u(-) defined by u(t) = T(t-s; A)f is the unique strongly continuous solution of (1.1) as long as/belongs to 3>(A).More generally, if A(t) depends on t in a suitably "smooth" manner, then equation (1.1) can be solved for an appropriate choice of the initial data/ The solution will be of the form u(t)=U(t,s)f U(t,s) is a bounded linear operator on 3£, U(t, s)U(s, r) = U(t, r), U(t, t) = Ifor O^ráiár, and [/(■, ■) is strongly continuous.The operator-valued function U(-, ■) is called an evolution operator or a generalized semigroup; the latter term will be used throughout the paper.We now state two of the main results of the paper.Theorem 1.1.Let A(-): [0, oo)^s/.Suppose that the operators T(s;A(t)) and T(a; A(t)) commute for all values oft, s, t, a. Suppose further there is a dense linear manifold@ in X such that Q>'<= ¡&(A(t)) for each r^O and lim A(t)f=A(r)f for eachfe 3¡ and each t ^ 0. Let è = O {{ Hm n ®(A(t))\ n ig : lim A(r)g = A(t)g\).igo \,(f->0+ |i-i|<e J