Geodesic
Variation of constants formula and exponential dichotomy for nonautonomous non-densely defined Cauchy problems
Canadian journal of mathematics, Tome 73 (2021) no. 5, pp. 1347-1389

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DOI

In this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.
DOI : 10.4153/S0008414X20000541
Mots-clés : Nonautonomous Cauchy problem, non-densely defined Cauchy problem, exponential dichotomy 
Magal, Pierre; Seydi, Ousmane. Variation of constants formula and exponential dichotomy for nonautonomous non-densely defined Cauchy problems. Canadian journal of mathematics, Tome 73 (2021) no. 5, pp. 1347-1389. doi: 10.4153/S0008414X20000541
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     title = {Variation of constants formula and exponential dichotomy for nonautonomous non-densely defined {Cauchy} problems},
     journal = {Canadian journal of mathematics},
     pages = {1347--1389},
     year = {2021},
     volume = {73},
     number = {5},
     doi = {10.4153/S0008414X20000541},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X20000541/}
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