Generalized exponentially bounded integrated semigroups

The subject of this paper is the analysis of sequences of infinitesimal generators and exponentially bounded integrated semigroups which are related to Cauchy problems(1)∂∂tu(t,x)−a(D)u(t,x)=f(t,x),u(0,x)=u0(x),t≥0,x∈Rd, with distributional initial data u0 and distributional right hand sides f through sequences of equations with regularized u0 and f, and a(D) approximated by a suitable sequences of (pseudo)differential operators an(D). Mainly, the paper deals with the comparison of sequences of infinitesimal generators and the determination of corresponding sequences of integrated semigroups. For this purpose, we introduce association, the relation of equivalence for infinitesimal generators on one side and the corresponding relations of equivalence of integrated semigroups on another side. The order of involved assumptions on generators essentially characterize the mutual dependence of sequences of infinitesimal generators and the corresponding sequences of integrated semigroups. Our motivation is presented in Section 3 where we show how Cauchy problem (1) with certain type of singularities is transferred to a sequence of Cauchy problems with classical solutions presented by one time integrated semigroups, and then discuss the solution to (1) in a weak sense (in the sense of distributions) or a very weak sense (when only a sequence of solutions exists without a weak limit). The effects of perturbations of infinitesimal generators to integrated semigroups is the core of the paper.