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Almost automorphic solutions for nonautonomous parabolic evolution equations

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Abstract

In this work, we study the existence and uniqueness of almost automorphic solutions for semilinear nonautonomous parabolic evolution equations with inhomogeneous boundary conditions using the exponential dichotomy. We assume that the homogeneous problem satisfies the “Acquistapace–Terreni” conditions and that the forcing terms are Stepanov-like almost automorphic. An example is given for illustration.

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References

  1. Acquistapace, P., Terreni, B.: A unified approach to abstract linear nonautonomous parabolic equations. Rend. Semin. Mat. Univ. Padova 78, 47–107 (1987)

    MathSciNet  MATH  Google Scholar 

  2. Acquistapace, P.: Evolution operators and strong solutions of abstract linear parabolic equations. Differ. Integral Equ. 1, 433–457 (1988)

    MathSciNet  MATH  Google Scholar 

  3. Akdad, A.N., Es-sebbar, B., Ezzinbi, K.: Composition theorems of Stepanov \(\mu \)-pseudo almost automorphic functions and applications to nonautonomous neutral evolution equations. Differ. Equ. Dyn. Syst. 25(3), 397–416 (2017)

    Article  MathSciNet  Google Scholar 

  4. Amann, H.: Linear and Quasilinear Parabolic Problems, Volume 1: Abstract Linear Theory. Birkhaüser, Basel (1995)

    Book  Google Scholar 

  5. Baroun, M., Boulite, S., N’Guérékata, G.M., Maniar, L.: Almost automorphy of parabolic evolution equations. Electron. J. Differ. Equ. 60, 1–9 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Baroun, M., Maniar, L., Schnaubelt, R.: Almost periodicity of parabolic evolution equations with inhomogeneous boundary values. Integral Equ. Oper. Theory 65, 169–193 (2009)

    Article  MathSciNet  Google Scholar 

  7. Baroun, M., Ezzinbi, K., Khalil, K., Maniar, L.: Pseudo almost periodic solutions for some parabolic evolution equations with Stepanov-like pseudo almost periodic forcing terms. J. Math. Anal. Appl. 462(1), 233–262 (2018)

    Article  MathSciNet  Google Scholar 

  8. Bochner, S.: Abstrakte fastperiodische funktionen. Acta Math. 61(1), 149–184 (1933)

    Article  MathSciNet  Google Scholar 

  9. Bochner, S.: A new approach to almost periodicity. Proc. Natl. Acad. Sci. USA 48(12), 2039–2043 (1962)

    Article  MathSciNet  Google Scholar 

  10. Bohr, H.: Zur theorie der fastperiodischen funktionen. Acta Math. 46(1–2), 101–214 (1925)

    Article  MathSciNet  Google Scholar 

  11. Blot, J., Cieutat, P., Ezzinbi, K.: Measure theory and pseudo almost automorphic functions: new developments and applications. Nonlinear Anal. 75, 2426–2447 (2012)

    Article  MathSciNet  Google Scholar 

  12. Blot, J., Cieutat, P., Ezzinbi, K.: New approach for weighted pseudo almost periodic functions under the light of measure theory, basic results and applications. Appl. Anal. 92(3), 493–526 (2013)

    Article  MathSciNet  Google Scholar 

  13. Diagana, T.: Weighted pseudo almost periodic functions and applications. C. R. Math. 343(10), 643–646 (2006)

    Article  MathSciNet  Google Scholar 

  14. Es-sebbar, B., Ezzinbi, K.: Almost periodicity and almost automorphy for some evolution equations using Favard’s theory in uniformly convex Banach spaces. Semigroup Forum 94(2), 229–259 (2017)

    Article  MathSciNet  Google Scholar 

  15. Es-sebbar, B., Ezzinbi, K.: Stepanov ergodic perturbations for some neutral partial functional differential equations. Math. Methods Appl. Sci. 39(8), 1945–1963 (2016)

    Article  MathSciNet  Google Scholar 

  16. Engel, R., Nagel, K.J.: One-Parameter Semigroups for Linear Evolution Equations, vol. 194. Springer, New York (2000)

    MATH  Google Scholar 

  17. Fan, Z., Liang, J., Xiao, T.J.: Composition of Stepanov-like pseudo almost automorphic functions and applications to nonautonomous evolution equations. Nonlinear Anal. Real World Appl. 13, 131–140 (2012)

    Article  MathSciNet  Google Scholar 

  18. Greiner, G.: Perturbing the boundary conditions of a generator. Houst. J. Math. 13, 213–229 (1987)

    MathSciNet  MATH  Google Scholar 

  19. Hong, J., Obaya, R., Sanz, A.: Almost periodic type solutions of some differential equations with piecewize constant argument. Nonlinear Anal. Theory Methods Appl. 45(6), 661–688 (2001)

    Article  Google Scholar 

  20. Lions, J.L.: Problèmes aux limites non homogènes et applications. Dunod, Paris (1968)

    MATH  Google Scholar 

  21. Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhauser, Basel (1995)

    Book  Google Scholar 

  22. Maniar, L., Schnaubelt, R.: Almost Periodicity of Inhomogeneous Parabolic Evolution Equations. Lecture Notes in Pure and Applied Mathematics, vol. 234, pp. 299–318. Dekker, New York (2003)

    Chapter  Google Scholar 

  23. N’Guérékata, G.M.: Topics in Almost Automorphy. Springer, New York (2005)

    MATH  Google Scholar 

  24. N’Guérékata, G.M., Pankov, A.: Stepanov-like almost automorphic functions and monotone evolution equations. Nonlinear Anal. 68, 2658–2667 (2008)

    Article  MathSciNet  Google Scholar 

  25. Schwartz, L.: Topologie Générale et Analyse Fonctionnelle. Hermann, Paris (1976). (in French)

    MATH  Google Scholar 

  26. Xiao, T.J., Liang, J., Zhang, J.: Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces. Semigroup Forum 76(3), 518–524 (2008)

    Article  MathSciNet  Google Scholar 

  27. Zhang, C.Y.: Pseudo almost periodic solutions of some differential equations. J. Math. Anal. Appl. 181(1), 62–76 (1994)

    Article  MathSciNet  Google Scholar 

  28. Zhang, R., Chang, Y.K., N’Guérékata, G.M.: New composition theorems of Stepanov-like weighted pseudo almost automorphic functions and applications to nonautonomous evolution equations. Nonlinear Anal. Real World Appl. 13, 2866–2879 (2012)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for careful reading of the manuscript and useful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, P.B. 2390-40000, Marrakesh, Morocco

    Mahmoud Baroun, Khalil Ezzinbi, Kamal Khalil & Lahcen Maniar

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  1. Mahmoud Baroun
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  2. Khalil Ezzinbi
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  3. Kamal Khalil
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  4. Lahcen Maniar
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Correspondence to Kamal Khalil.

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Communicated by Abdelaziz Rhandi.

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Baroun, M., Ezzinbi, K., Khalil, K. et al. Almost automorphic solutions for nonautonomous parabolic evolution equations. Semigroup Forum 99, 525–567 (2019). https://doi.org/10.1007/s00233-019-10045-w

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