Exponential Characteristic Of Nonautonomous Systems In Banach Spaces

This paper is concerned with the exponential characteristic of the followingnonautonomous system in Banach space X where (A (t), D (A (t)))are unbounded linear operators in X and f is an X valuedfunction on the interval [0,∞). The exponential characteristic of a nonautonomoussystem with uniformly bounded norms and periodic operator coefficient in Banachspaces, and that of a nonautonomous system with uniformly bounded norms butnonperiodic operator coefficient are determined in [10]and [12], respectively. Basedon the work of these two papers, for the unbounded linear operator A (t)whichgenerates an evolution system, with the aid of the differentiability and theexponential growth of the evolution system, results of a kind of nonautonomoussystem with an unbounded nonperiodic operator coefficient in parabolic case inBanach spaces are obtained.This paper are divided into three chapters. In Chapter1, we firstly give a briefreview of the background of semigroups of linear operators, development ofnonautonomous system in Banach spaces, and status quo of researches aboutexponential characteristic estimation at home and abroad. Finally, a brief introductionto the main result of this paper is presented as well.The second chapter provides some preliminary knowledge of nonautonomoussystem in Banach spaces and theorems which will be cited in the latter of this paper.In this way, some properties of nonautonomous system with zero initial-value condition are obtained.The third chapter focuses on setting forth the key work of this paper, i.e.,theconclusion and its proof of the exponential characteristic of a nonautonomoussystem in Banach spaces. Four steps are carried out in our approach.