Parabolic evolution equations with asymptotically autonomous delay

Abstract

We study retarded parabolic non-autonomous evolution equations whose coefficients converge as t → ∞ t\to \infty , 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.