Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

Editor-in-Chief: Diagana, Toka

Managing Editor: Cánovas, Jose


Mathematical Citation Quotient (MCQ) 2017: 0.71

Open Access
Online
ISSN
2353-0626
See all formats and pricing
More options …

Nonautonomous partial functional differential equations; existence and regularity

Moussa El-Khalil Kpoumiè
  • Corresponding author
  • Université de Ngaoundéré, École de Géologie et d’Exploitation Minière, Département de Mathématiques Appliquées et Informatique, B.P. 115 Meiganga, Cameroon
  • Université de Yaoundé I, Faculté des Sciences, Departement de Mathematiques, B.P. 812 Yaoundé, Cameroon
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Khalil Ezzinbi
  • Université Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques, B.P. 2390, Marrakesh, Morocco
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ David Békollè
Published Online: 2017-12-07 | DOI: https://doi.org/10.1515/msds-2017-0010

Abstract

The aim of this work is to establish several results on the existence and regularity of solutions for some nondensely nonautonomous partial functional differential equations with finite delay in a Banach space. We assume that the linear part is not necessarily densely defined and generates an evolution family under the conditions introduced by N. Tanaka.We show the local existence of the mild solutions which may blow up at the finite time. Secondly,we give sufficient conditions ensuring the existence of the strict solutions. Finally, we consider a reaction diffusion equation with delay to illustrate the obtained results.

Keywords: Nonautonomous equation; evolution family; generalized variation of constants formula; compatibility conditions; stability conditions; mild and strict solutions

References

  • [1] P. Acquistapace and B. Terreni. A unified approach to abstract linear nonautonomous parabolic equations. Rendiconti del Seminario Matematico della Università di Padova, 78:47-107, (1987).Google Scholar

  • [2] M. Adimy and K. Ezzinbi. Existence, regularity, stability and boundedness for some partial functional differential equations. In Séminaires et Congrès, number 17, pages 157-190, (2009).Google Scholar

  • [3] W. Arendt. Vector-valued laplace transforms and cauchy problems. Israel Journal of Mathematics, 59(3), (1987).Google Scholar

  • [4] M. G. Crandall and A. Pazy. Nonlinear evolution equations in banach spaces. Israel J. Math., 11:57-94, (1972).Google Scholar

  • [5] Sever Silvestru Dragomir. Some Gronwall Type Inequalities and Applications. Nova Science Pub Incorporated, 2003.Google Scholar

  • [6] A. G. Kartsatos and K.-Y. Shin. Solvability of functional evolutions via compactness methods in general banach spaces. Nonlmeor Analysis, Theory, Methods and Applicolions, 21(7):517-535, (1993).Google Scholar

  • [7] T. Kato. Linear evolution equations of hyperbolic type. Journal of the Faculty of Science, University of Tokyo, 17:241-258, (1970).Google Scholar

  • [8] T. Kato. Linear evolution equations of hyperbolic type ii. J. Math. Soc. Japan, 25(4):648-666, (1973).Google Scholar

  • [9] H. Kellerman and M. Hieber. Integrated semigroups. Journal of Functional Analysis, 84:160-180, (1989).CrossrefGoogle Scholar

  • [10] M. Laklach M. Adimy and K. Ezzinbi. Nonlinear semigroup of a class of abstract semilinear functional differential equations with a non-dense domain. Acta Mathematica Sinica, English Series, 20:933-942, (2004).Google Scholar

  • [11] E. Mitidieri and L. I. Vrabie. Existence for nonlinear functional differential equations. HiroshimaMath. J., 17:627-649, (1987).Web of ScienceGoogle Scholar

  • [12] H. Oka and N. Tanaka. Evolution operators generated by non-densely defined operators. Math. Nachr., 278(11):1285-1296, August (2005).Google Scholar

  • [13] N. H. Pavel. Nonlinear Evolution Operators and Semigroups, Lecture Notes Math., volume 1260. Springer, Berlin, 1987.Google Scholar

  • [14] A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer Verlang, New York, 1983.Google Scholar

  • [15] G. Da Prato and E. Sinestradi. Non autonomous evolution operators of hyperbolic type. Semigroup Forum, 45:302-321, (1992).CrossrefGoogle Scholar

  • [16] W. M. Ruess. Existence of solutions to partial functional differential equations with delay. In A. G. Kartsatos, editor, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, volume 178, page 259-288. 1996.Google Scholar

  • [17] H. Tanabe. Equations of Evolutions. Pitman, London, 1979.Google Scholar

  • [18] N. Tanaka. Semilinear equations in the hyperbolic case. Nonlinear Analysis, Theory, Methods and Application, 24(5):773- 788, March (1995).Google Scholar

  • [19] N. Tanaka. Quasilinear evolution equationswith non-densely defined operators. Differential and Integral Equations, 5:1067- 1106, (1996).Google Scholar

  • [20] N. Tanaka. Generation of linear evolution operators. Proceedings of the AmericanMathematical Society, 128(7):2007-2015, November 24 (1999).Google Scholar

  • [21] C. C. Travis and G. F. Webb. Existence and stability for partial functional differential equations. Transactions of the American Mathematical Society, 200:395-418, (1974).Google Scholar

  • [22] J. Wu. Theory and Applications of Partial Functional Differential Equations. Springer Verlang, New york, 1996.Google Scholar

About the article

Received: 2017-04-29

Accepted: 2017-11-08

Published Online: 2017-12-07

Published in Print: 2017-11-27


Citation Information: Nonautonomous Dynamical Systems, Volume 4, Issue 1, Pages 108–127, ISSN (Online) 2353-0626, DOI: https://doi.org/10.1515/msds-2017-0010.

Export Citation

© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in