In this paper, we are interested in the wellposedness of inhomogeneous nonautonomous boundary Cauchy problems. We prove variation of constants formulas and Dyson-Phillips series which will allow us to study the asymptotic behaviour of the solutions to these problems. 1991 Mathematics Subject Classification: 34B05, 35B15, 35B20, 35B35.8 > < > : ð1:1Þwhere J :¼ R or R þ , A max ðtÞ is an unbounded operator on a Banach space X endowed with a maximal domain DðA max ðtÞÞ, and LðtÞ : DðA max ðtÞÞ ! qX , FðtÞ : X ! qX , t A J, with qX is a 'boundary space'. These abstract problems model natural phenomena such as population equations, retarded di¤erential (di¤erence) equations and boundary control problems. In the autonomous case, their study was started by Greiner [21,22,23] using the perturbation of the domains of semigroups, and Desch-This work is finished while the second author was visiting Tü bingen University. He would like to thank Rainer Nagel and Roland Schnaubelt for many useful discussions and comments, and he is grateful for the financial support of the Alexander von Humboldt Foundation. The third author acknowledges the partial support of ''Centre International de Systèmes Dynamiques (CISD)'' at Caddi Ayyad University Marrakesh. Brought to you by | Tulane University Authenticated Download Date | 1/4/15 12:21 AM Brought to you by | Tulane University Authenticated Download Date | 1/4/15 12:21 AM Brought to you by | Tulane University Authenticated Download Date | 1/4/15 12:21 AM

In this paper we prove the existence and the uniqueness of the classical solution of non-autonomous inhomogeneous boundary Cauchy problems, and that this solution is given by a variation of constants formula. This result is applied to show the existence of solutions of a retarded equation.

In this paper we present results concerning the existence, stability and local attractivity for non-autonomous semilinear boundary Cauchy problems. In our method, we assume certain smoothness properties on the linear part and the local lipshitz continuity on the nonlinear perturbation. We apply our abstract results to population equations.

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