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Asymptotic behaviour of parabolic nonautonomous evolution equations. (English) Zbl 1082.35092
Iannelli, Mimmo (ed.) et al., Functional analytic methods for evolution equations. Based on lectures given at the autumn school on evolution equations and semigroups, Levico Terme, Trento, Italy, October 28–November 2, 2001. Berlin: Springer (ISBN 3-540-23030-0/pbk). Lecture Notes in Mathematics 1855, 401-472 (2004).

The paper is concerned with the Cauchy problem for the nonautonomous evolution equation

$\frac{d}{dt}u\left(t\right)=A\left(t\right)u\left(t\right)+f\left(t\right),\phantom{\rule{4pt}{0ex}}t>s,\phantom{\rule{1.em}{0ex}}\phantom{\rule{4pt}{0ex}}u\left(s\right)=x,\phantom{\rule{1.em}{0ex}}t,s\in J\subset \left\{\left[a,\infty \right),ℝ\right\},$

where $A\left(t\right),t\in J$, are linear operators on a Banach space $X$ and $x\in X,f\in C\left(J,X\right)$. Solutions to the problem and their properties are under consideration. A self-contained and systematic presentation of known results as well as authors’s new results are given.

The paper consists of six sections: 1. Introduction 2. Parabolic Evolution Equation 3. Exponential Dichotomy 4. Exponential Dichotomy of Parabolic Evolution Equations 5. Inhomogeneous Problems 6. Convergent Solutions for a Quasilinear Equation.

The main attention is given to the problem of exponential dichotomy and to the problem of the exponential dichotomy persistence under ‘small’ perturbations. The complete list of bibliography is represented.

MSC:
 35K90 Abstract parabolic equations 35B40 Asymptotic behavior of solutions of PDE
Keywords:
exponential dichotomy; splitting