Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time

We establish maximal regularity of type LP for a parabolic evolution equati on u'(t) = A(t)u(t) + f(t) with A(.) is an element of C([0, T], L(D(A(0)), X)) and construct the corresponding evolution family on the underlying Bana ch space X. Our proofs are based on the operator sum method and the use of evolution semigroups. The results are applied to parabolic partial differen tial equations with continuous coefficients. (C) 2001 Academic Press.