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Wellposedness and asymptotic behaviour of nonautonomous linear evolution equations
 Progr. Nonlinear Differential Equations Appl
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Parabolic Evolution Equations With Asymptotically Autonomous Delay
 Report No.2, Fachbereich Mathematik und Informatik, Universitat
, 2001
"... . We study retarded parabolic nonautonomous evolution equations whose coefficients converge as t ! 1 such that the autonomous problem in the limit has an exponential dichotomy. Then the nonautonomous problem inherits the exponential dichotomy and the solution of the inhomogeneous equation ten ..."
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. We study retarded parabolic nonautonomous evolution equations whose coefficients converge as t ! 1 such that the autonomous problem in the limit has an exponential dichotomy. Then the nonautonomous 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 longterm 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 `AcquistapaceTerreni' condition (see (P) below) and that there exists another sectorial operator A such that lim t!1 R(w; A(t...
ftp ejde.math.swt.edu (login: ftp) NONAUTONOMOUS RETARDED DIFFERENTIAL EQUATIONS: THE VARIATION OF CONSTANTS FORMULAS AND THE ASYMPTOTIC BEHAVIOUR
"... Abstract. This paper is devoted to show a variation of constants formula for the operator solution to the nonautonomous retarded differential equation x′(t) = A(t)x(t) + L(t)xt + f(t), xs = ϕ, t ≥ s ≥ 0, in terms of the inhomogeneous term f, which will allow us to study the asymptotic behaviour o ..."
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Abstract. This paper is devoted to show a variation of constants formula for the operator solution to the nonautonomous retarded differential equation x′(t) = A(t)x(t) + L(t)xt + f(t), xs = ϕ, t ≥ s ≥ 0, in terms of the inhomogeneous term f, which will allow us to study the asymptotic behaviour of this solution. We treat also the existence of fundamental solutions and the stability of semilinear retarded differential equations. 1.
ftp ejde.math.txstate.edu (login: ftp) ALMOST AUTOMORPHY OF SEMILINEAR PARABOLIC EVOLUTION EQUATIONS
"... Abstract. This paper studies the existence and uniqueness of almost automorphic mild solutions to the semilinear parabolic evolution equation u′(t) = A(t)u(t) + f(t, u(t)), assuming that the linear operators A(·) satisfy the ’Acquistapace–Terreni ’ conditions, the evolution family generated by A( ..."
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Abstract. This paper studies the existence and uniqueness of almost automorphic mild solutions to the semilinear parabolic evolution equation u′(t) = A(t)u(t) + f(t, u(t)), assuming that the linear operators A(·) satisfy the ’Acquistapace–Terreni ’ conditions, the evolution family generated by A(·) has an exponential dichotomy, and the resolvent R(ω,A(·)), and f are almost automorphic. 1.
Von der Fakultät für Mathematik und Physik der Gottfried Wilhelm Leibniz Universität Hannover
, 2006
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