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Variation of constants formula and exponential dichotomy for nonautonomous non-densely defined Cauchy problems

Part of: Groups and semigroups of linear operators, their generalizations and applications Stability theory Topological dynamics

Published online by Cambridge University Press:  29 June 2020

Pierre Magal  and
Ousmane Seydi
Pierre Magal*
Affiliation:
Université de Bordeaux, IMB, UMR 5251, F-33076Bordeaux, France and CNRS, IMB, UMR 5251, F-33400Talence, France
Ousmane Seydi
Affiliation:
Département Tronc Commun, Ecole Polytechnique de Thiès, Thiès21001, Sénégal e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

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.

Keywords

Nonautonomous Cauchy problemnon-densely defined Cauchy problemexponential dichotomy

MSC classification

Primary: 37B55: Nonautonomous dynamical systems
Secondary: 47D62: Integrated semigroups 34D09: Dichotomy, trichotomy
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 5 , October 2021 , pp. 1347 - 1389
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Copyright
© Canadian Mathematical Society 2020

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References

Arendt, W., Resolvent positive operators. Proc. Lond. Math. Soc. 54(1987), 321–349. http://dx.doi.org/10.1112/plms/s3-54.2.321 CrossRefGoogle Scholar
Arendt, W., Vector valued Laplace transforms and Cauchy problems. Israel J. Math. 59(1987), 327–352. http://dx.doi.org/10.1007/BF02774144 CrossRefGoogle Scholar
Baskakov, A. G., Semigroups of difference operators in spectral analysis of linear differential operators. Funct. Anal. Appl. 30(1996), 149–157. http://dx.doi.org/10.1007/BF02509501 CrossRefGoogle Scholar
Boulite, S., Maniar, L., and Moussi, M., Non-autonomous retarded differential equations: variation of constants formulas and asymptotic behaviour . Elect. J. Differ. Equat. (2003), no. 62, 1–15.Google Scholar
Boulite, S., Maniar, L., and Moussi, M.,Wellposedness and asymptotic behaviour of non-autonomous boundary Cauchy problems. Forum Math. 18(2006), 611–638. http://dx.doi.org/10.1015/FORUM.2006.032 CrossRefGoogle Scholar
Ducrot, A., Magal, P., and Prevost, K., Integrated semigroups and parabolic equations. Part I: linear perturbation of almost sectorial operators. J. Evol. Equat. 10(2010), 263–291. http://dx.doi.org/10.1007/s00028-009-0049-z CrossRefGoogle Scholar
Ducrot, A., Magal, P., and Seydi, O., A finite-time condition for exponential trichotomy in infinite dynamical systems. Canad. J. Math. 67(2015), 1065–1090. http://dx.doi.org/10.4153/CJM-2014-023-3 CrossRefGoogle Scholar
Ducrot, A., Magal, P., and Seydi, O., Persistence of exponential trichotomy for linear operators: A Lyapunov-Perron approach. J. Dyn. Differ. Equat. 28(2016), 93–126. http://dx.doi.org/10.1007/s10884-015-9493-3 CrossRefGoogle Scholar
Ducrot, A., Liu, Z., and Magal, P., Projectors on the generalized eigenspaces for neutral functional differential equations in L spaces. Canad. J. Math. 62(2010), 74–93. http://dx.doi.org/10.4153/CJM-2010-005-2 CrossRefGoogle Scholar
Engel, K.-J. and Nagel, R., One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.Google Scholar
Gühring, G. and Räbiger, F., Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations . Abstr. App. Anal. 4(1999), 169–194. http://dx.doi.org/10.1155/S1085337599000214 CrossRefGoogle Scholar
Hale, J. K. and Lin, X. B., Heteroclinic orbits for retarded functional differential equations . J. Differ. Equat. 65(1986), 175–202. http://dx.doi.org/10.1016/0022-0396(86)90032-X CrossRefGoogle Scholar
Henry, D., Geometric theory of semilinear parabolic equations . Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.Google Scholar
Kellermann, H. and Hieber, M., Integrated semigroups . J. Funct. Anal. 84(1989), 160–180. http://dx.doi.org/10.1016/0022-1236(89)90116-X CrossRefGoogle Scholar
Latushkin, Y., Randolph, T., and Schnaubelt, R., Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces . J. Dyn. Differ. Equat. 10(1998), 489–510. http://dx.doi.org/10.1023/A:1022609414870 CrossRefGoogle Scholar
Levitan, B. M. and Zhikov, V. V., Almost periodic functions and differential equations. Translated from the Russian by Longdon, L. W., Cambridge University Press, Cambridge-New York, 1982.Google Scholar
Lian, Z. and Lu, K., Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space . Mem. Amer. Math. Soc. 206(2010), no. 967. http://dx.doi.org/10.1090/S0065-9266-10-00574-0 Google Scholar
Magal, P. and Ruan, S., On integrated semigroups and age structured models in L ${}^p$ spaces. Differ. Integral Equat. 20(2007), 197–239.Google Scholar
Magal, P. and Ruan, S., On semilinear Cauchy problems with non-dense domain . Adv. Differ. Equat. 14(2009), 1041–1084.Google Scholar
Magal, P. and Ruan, S., Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models . Mem. Amer. Math. Soc. 202(2009), no. 951. http://dx.doi.org/10.1090/S0065-9266-09-00568-7 Google Scholar
Schnaubelt, R., Sufficient conditions for exponential stability and dichotomy of evolution equations . Forum Math. 11(1999), no. 5, 543–566. http://dx.doi.org/10.1515/form.1999.013 CrossRefGoogle Scholar
Thieme, H. R., Integrated semigroups and integrated solutions to abstract Cauchy problems . J. Math. Anal. App. 152(1990), 416–447. http://dx.doi.org/10.1016/0022-247X(90)90074-P CrossRefGoogle Scholar
Thieme, H. R., Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem. J. Evol. Equat. 8(2008), 283–305. http://dx.doi.org/10.1007/s00028-007-0355-2CrossRefGoogle Scholar