# Generalized exponentially bounded integrated semigroups

## TL;DR

This paper studies sequences of exponentially bounded integrated semigroups related to Cauchy problems with distributional data, introducing concepts of association and equivalence to analyze their generators and semigroups.

## Contribution

It introduces the notions of association and equivalence for infinitesimal generators and integrated semigroups, providing a framework to analyze their mutual dependence in regularized Cauchy problems.

## Key findings

- Established criteria for the equivalence of generators and semigroups.
- Characterized the dependence of integrated semigroups on their generators.
- Provided a theoretical foundation for analyzing regularized PDE solutions.

## Abstract

The main subject of this paper is the analysis of sequences of exponentially bounded integrated semigroups which are related to Cauchy problems \begin{equation}\label{jed} \frac{\partial}{\partial t}u(t,x)-a(D)u(t,x)=f(t,x), \quad u(0,x)=u_0(x), \quad t\geq 0, \ x\in \mathbb R^d, \end{equation} with a distributional initial data $u_0$ and a distributional right hand side $f$ through a sequence of equations with regularized $u_0$ and $f$ and a sequence of (pseudo) differential operators $a_n(D)$ instead of $a(D)$. Comparison of sequences of infinitesimal generators and the determination of corresponding sequences of integrated semigroups are the main subject of the paper. 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.