Solution Estimates for Semilinear Non-autonomous Evolution Equations with Differentiable Linear Parts

In a Hilbert space we consider the equation du(t)/dt=A(t)u(t)+F(t,u(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$du(t)/dt=A(t)u(t)+F(t,u(t))$$\end{document}, where A(t) is an unbounded operator, having a bounded strong derivative, and F is a continuous mapping. We derive norm estimates for solutions of the considered equation. These estimates give us the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-stability and absolute stability conditions. To the best of our knowledge an absolute stability test for nonautonomous evolution equations has been obtained for the first time. Our main tool is the norm estimate for the derivative of a solution of the time-dependent Lyapunov equation.