Evolution equations governed by families of weighted operators

Abstract

In order to develop a Lebesgue approach for the fully non-linear non autonomous evolution problem, CPAα={dudt+α(t)Aα(t)u∍0} with tϵI ⊆ [0,T], in an arbitrary Banach space X, we define an abstract L1 - comparison mode (called coherence) between multivalued time dependent families of operators (Aα(s))sϵI and (Bβ(t))tϵJ defined on compact subintervals I and J of [0,T] and weighted by functions α and β which belong to L∞([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 Bénilan'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, Bénilan, 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∞ -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], …).