Skip to main content
SpringerLink
Log in

Semilinear Delay Evolution Equations with Nonlocal Initial Conditions

  • Chapter
  • First Online:
New Prospects in Direct, Inverse and Control Problems for Evolution Equations

Part of the book series: Springer INdAM Series ((SINDAMS,volume 10))

Abstract

An existence and asymptotic behaviour result for a class of semilinear delay evolution equations subjected to nonlocal initial conditions is established. An application to a semilinear wave equation is also discussed.

In memory of Professor Alfredo Lorenzi

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aizicovici, S., Lee, H.: Nonlinear nonlocal Cauchy problems in Banach spaces. Appl. Math. Lett. 18, 401–407 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aizicovici, S., McKibben, M.: Existence results for a class of abstract nonlocal Cauchy problems. Nonlinear Anal. 39, 649–668 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Avalishvili, G., Avalishvili, M.: Nonclassical problems with nonlocal initial conditions for abstract second-order evolution equations. Bull. Georgian Natl. Acad. Sci. (N.S.) 5, 17–24 (2011)

    Google Scholar 

  4. Becker, R.I.: Periodic solutions of semilinear equations of evolution of compact type. J. Math. Anal. Appl. 82, 33–48 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  5. Burlică, M.D., Roşu, D.: A class of nonlinear delay evolution equations with nonlocal initial conditions. Proc. Am. Math. Soc. 142, 2445–2458 (2014)

    Article  MATH  Google Scholar 

  6. Burlică, M., Roşu, D., Vrabie, I.I.: Continuity with respect to the data for a delay evolution equation with nonlocal initial conditions. Libertas Math. (New series) 32, 37–48 (2012)

    Google Scholar 

  7. Burlică, M.D., Roşu, D., Vrabie, I.I.: Abstract reaction-diffusion systems with nonlocal initial conditions. Nonlinear Anal. 94, 107–119 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  8. Byszewski, L.: Theorems about the existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problems. J. Math. Anal. Appl. 162, 494–505 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cârjă, O., Necula, M., Vrabie, I.I.: Viability, Invariance and Applications. Mathematics Studies, vol. 207. Elsevier, North-Holland (2007)

    Google Scholar 

  10. Deng, K.: Exponential decay of solutions of semilinear parabolic equations with initial boundary conditions. J. Math. Anal. Appl. 179, 630–637 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Di Blasio, G., Lorenzi, A.: Identification problems for integro-differential delay equations. Differ. Integral Equ. 16, 1385–1408 (2003)

    MATH  Google Scholar 

  12. García-Falset, J., Reich, S.: Integral solutions to a class of nonlocal evolution equations. Commun. Contemp. Math. 12, 1032–1054 (2010)

    Article  Google Scholar 

  13. Gordeziani, D.G.: On some initial conditions for parabolic equations. Rep. Enlarged Sess. Sem. I. Vekua Inst. Appl. Math. 4, 57–60 (1989)

    Google Scholar 

  14. Gordeziani, D.G., Avalishvili, M., Avalishvili, G.: On the investigation of one nonclassical problem for Navier–Stokes equations. AMI 7, 66–77 (2002)

    MATH  MathSciNet  Google Scholar 

  15. Hale, J.: Functional Differential Equations. Applied Mathematical Sciences, vol. 3. Springer, New York (1971)

    Google Scholar 

  16. Lorenzi, A., Vrabie, I.I.: Identification of a source term in a semilinear evolution delay equation. An. Ştiint. Univ. “Al. I. Cuza” din Iaşi (S.N.), Matematică LXI, 1–39 (2015)

    Google Scholar 

  17. Lorenzi, A., Vrabie, I.I.: An identification problem for a semilinear evolution delay equation. J. Inverse Ill-Posed Probl. 22, 209–244 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  18. McKibben, M.: Discovering Evolution Equations with Applications, I. Deterministic Models. Appl. Math. Nonlinear Sci. Ser. Chapman & Hall/CRC, Boca Raton (2011)

    Book  Google Scholar 

  19. Mitidieri, E., Vrabie, I.I.: Existence for nonlinear functional differential equations. Hiroshima Math. J. 17, 627–649 (1987)

    MATH  MathSciNet  Google Scholar 

  20. Mitidieri, E., Vrabie, I.I.: A class of strongly nonlinear functional differential equations. Ann. Mat. Pura Appl. 4-CLI, 125–147 (1988)

    Google Scholar 

  21. Olmstead, W.E., Roberts, C.A.: The one-dimensional heat equation with a nonlocal initial condition. Appl. Math. Lett. 10, 89–94 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  22. Paicu, A., Vrabie, I.I.: A class of nonlinear evolution equations subjected to nonlocal initial conditions. Nonlinear Anal. 72, 4091–4100 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. Rabier, F., Courtier, P., Ehrendorfer, M.: Four-dimensional data assimilation: comparison of variational and sequential algorithms. Q. J. R. Meteorol. Sci. 118, 673–713 (1992)

    Article  Google Scholar 

  24. Schaefer, H.: Über die Methode der a priori-Schranken. Math. Ann. 129, 415–416 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  25. Shelukhin, V.V.: A nonlocal in time model for radionuclides propagation in stokes fluid, dynamics of fluids with free boundaries. Inst. Hydrodynam. 107, 180–193 (1993)

    MATH  MathSciNet  Google Scholar 

  26. Shelukhin, V.V.: A problem nonlocal in time for the equations of the dynamics of a barotropic ocean. Siberian Math. J. 36, 701–724 (1995)

    Article  MathSciNet  Google Scholar 

  27. Vrabie, I.I.: C 0-semigroups and applications. North-Holland, Amsterdam (2003)

    MATH  Google Scholar 

  28. Vrabie, I.I.: Existence for nonlinear evolution inclusions with nonlocal retarded initial conditions. Nonlinear Anal. 74, 7047–7060 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Vrabie, I.I.: Existence in the large for nonlinear delay evolution inclusions with nonlocal initial conditions. J. Funct. Anal. 262, 1363–1391 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  30. Vrabie, I.I.: Nonlinear retarded evolution equations with nonlocal initial conditions. Dyn. Syst. Appl. 21, 417–440 (2012)

    MATH  MathSciNet  Google Scholar 

  31. Vrabie, I.I.: Global solutions for nonlinear delay evolution inclusions with nonlocal initial conditions. Set-valued Var. Anal. 20, 477–497 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  32. Vrabie, I.I.: Almost periodic solutions for nonlinear delay evolutions with nonlocal initial conditions. J. Evol. Equ. 13, 693–714 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  33. Vrabie, I.I.: Delay evolution equations with mixed nonlocal plus local initial conditions. Commun. Contemp. Math. (2013). doi: 10.1142/S0219199713500351.

    MATH  Google Scholar 

Download references

Acknowledgements

(a) This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS–UEFISCDI, project number PN-II-ID-PCE-2011-3-0052.

(b)The author expresses his warmest thanks to the referee for the very careful reading of the paper and for his/her useful suggestions and remarks.

Author information

Authors and Affiliations

  1. Faculty of Mathematics Al. Cuza University, Iaşi Romania and Octav Mayer Institute of Mathematics (Romanian Academy), Iaşi, Romania

    Ioan I. Vrabie

Authors
  1. Ioan I. Vrabie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ioan I. Vrabie .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Bologna, Milano, Italy

    Angelo Favini

  2. Dipartimento di Matematica, Università di Bari, Bari, Bari, Italy

    Genni Fragnelli

  3. Dipartimento di Matematica, Università di Bari, Bari, Bari, Italy

    Rosa Maria Mininni

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Vrabie, I.I. (2014). Semilinear Delay Evolution Equations with Nonlocal Initial Conditions. In: Favini, A., Fragnelli, G., Mininni, R. (eds) New Prospects in Direct, Inverse and Control Problems for Evolution Equations. Springer INdAM Series, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-11406-4_19

Download citation

Publish with us

Policies and ethics

Search

Navigation