Stability for non-autonomous linear evolution equations with Lp-maximal regularity

We study stability and integrability of linear non-autonomous evolutionary Cauchy-problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P),\left\{ \begin{gathered} \dot u(t) + A(t)u(t) = f(t) t - a.e. on [0,\tau ] \hfill \\ u(0) = 0, \hfill \\ \end{gathered} \right.$$\end{document} where A: [0, τ] → L(X,D) is a bounded and strongly measurable function and X, D are Banach spaces such that [inline-graphic not available: see fulltext]. Our main concern is to characterize Lp-maximal regularity and to give an explicit approximation of the problem (P).