## Parabolic Evolution Equations With Asymptotically Autonomous Delay (2001)

Venue: | Report No.2, Fachbereich Mathematik und Informatik, Universitat |

Citations: | 3 - 3 self |

### BibTeX

@TECHREPORT{Schnaubelt01parabolicevolution,

author = {Roland Schnaubelt},

title = {Parabolic Evolution Equations With Asymptotically Autonomous Delay},

institution = {Report No.2, Fachbereich Mathematik und Informatik, Universitat},

year = {2001}

}

### OpenURL

### Abstract

. We study retarded parabolic non--autonomous evolution equations whose coefficients converge as t ! 1 such that the autonomous problem in the limit has an exponential dichotomy. Then the non--autonomous problem inherits the exponential dichotomy and the solution of the inhomogeneous equation tends to the stationary solution at infinity. We use a generalized characteristic equation to deduce the exponential dichotomy and new representation formulas for the solution of the inhomogeneous equation. 1. Introduction In the present paper we continue the investigation of the long--term behaviour of asymptotically autonomous evolution equations begun in [30]. There we studied the Cauchy problem u(t) = A(t)u(t) + f(t); t ? s 0; u(s) = x; (1.1) on a Banach space X assuming that the linear operators A(t), t 0, are sectorial of the same type and satisfy the `Acquistapace--Terreni' condition (see (P) below) and that there exists another sectorial operator A such that lim t!1 R(w; A(t...