Evolution equations governed by families of weighted operators

In order to develop a Lebesgue approach for the fully non-linear non autonomous evolution problem, CPAalpha = {du/dt + alpha(t)A(alpha)(t)u There Exists 0} with t is an element of I subset of or equal to [0,T], in an arbitrary Banach space X, we define an abstract L-1- comparison mode (called coherence) between multivalued time dependent families of operators (A(alpha)(s))(s is an element of I) and (B-beta(t))(t is an element of J) defined on compact subintervals I and J of [0,T] and weighted by functions alpha and beta which belong to L-infinity([0,T]; R+). The solutions of these problems are limit of discrete schemes and the crucial point is to define these approximations in a Lebesgue sense. The results about this Cauchy problem consist in existence of an evolution operator, integral inequalities (extending Benilan's inequalities for integral solutions), and continuous properties; they extend the theory of evolution equations initiated at the beginning of the seventeenth by Crandall, Liggett, Benilan, Kobayashi, Evans, ([10], [12],...), and include more recent generalizations as in [18] and [6]. This general study motivated by the observation problem of a heat exchanger (see [16]) where a L-infinity-control multiplies an unbounded operator, establishes in Theorem 3.4 a suitable continuity property with respect to the weak* topology on the weights (see applications in [3], [7], [20],...). (C) Elsevier, Paris.