Skip to main content
SpringerLink
Log in

Existence of solutions of the nonautonomous abstract Cauchy problem of second order

  • RESEARCH ARTICLE
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

In this paper the existence of solutions of a nonautonomous abstract Cauchy problem of second order is considered. Assuming appropriate conditions on the operator of the equation, we establish the existence of mild solutions and, in some cases, we construct an evolution operator associated to the homogeneous equation. Using this evolution operator we obtain existence of solutions for the inhomogeneous equation. Finally, we apply our results to study the existence of solutions of the nonautonomous wave equation.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arosio, A.: Abstract linear hyperbolic equations with variable domain. Ann. Mat. Pura Appl. 135, 173–218 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bales, L.A.: Higher order single step fully discrete approximations for second order hyperbolic equations with time dependent coefficients. SIAM J. Numer. Anal. 23, 27–43 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bales, L.A.: Higher order single step fully discrete approximations for nonlinear second order hyperbolic equations. Comput. Math. Appl. 12, 581–604 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bales, L.A., Dougalis, V.A., Serbin, S.M.: Cosine methods for second-order hyperbolic equations with time-dependent coefficients. Math. Comput. 45, 65–89 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  5. Batty, C.J.K., Chill, R., Srivastava, S.: Maximal regularity for second order non-autonomous Cauchy problems. Stud. Math. 189, 205–223 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bochenek, J.: Existence of the fundamental solution of a second order evolution equation. Ann. Pol. Math. 66, 15–35 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Cioranescu, I., Ubilla, P.: A characterization of uniformly bounded cosine functions generators. Integral Equ. Oper. Theory 12, 1–11 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Faraci, F., Iannizzotto, A.: A multiplicity theorem for a perturbed second-order non-autonomous system. Proc. Edinb. Math. Soc. 49, 267–275 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fattorini, H.O.: Second Order Linear Differential Equations in Banach Spaces. North-Holland Mathematics Studies, vol. 108. North-Holland, Amsterdam (1985)

    MATH  Google Scholar 

  10. Favini, A., Gal, C.G., Ruiz Goldstein, G., Goldstein, J.A., Romanelli, S.: The non-autonomous wave equation with general Wentzell boundary conditions. Proc. R. Soc. Edinb. 135A, 317–329 (2005)

    MathSciNet  Google Scholar 

  11. Goldstein, J.A.: Time dependent hyperbolic equations. J. Funct. Anal. 4, 31–49 (1969)

    Article  MATH  Google Scholar 

  12. Goldstein, J.A.: A perturbation theory for evolution equations and some applications. Ill. J. Math. 18, 196–207 (1974)

    MATH  Google Scholar 

  13. Goldstein, J.A.: Variable domain second order evolution equations. Appl. Anal. 5, 283–291 (1976)

    Article  MATH  Google Scholar 

  14. Ha, J., Nakagiri, S.-I., Tanabe, H.: Gateaux differentiability of solution mappings for semilinear second-order evolution equations. J. Math. Anal. Appl. 310, 518–532 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ha, J., Nakagiri, S.-I., Tanabe, H.: Frechet differentiability of solution mappings for semilinear second order evolution equations. J. Math. Anal. Appl. 346, 374–383 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Haase, M.: The Functional Calculus for Sectorial Operators. Birkhäuser, Basel (2006)

    Book  MATH  Google Scholar 

  17. Henríquez, H.R., Vásquez, C.: Differentiability of the second order abstract Cauchy problem. Semigroup Forum 64, 472–488 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kato, T.: Integration of the equation of evolution in a Banach space. J. Math. Soc. Jpn. 5, 208–234 (1953)

    Article  MATH  Google Scholar 

  19. Kato, T.: On linear differential equations in Banach spaces. Commun. Pure Appl. Math. 9, 479–486 (1956)

    Article  MATH  Google Scholar 

  20. Kato, T.: Linear evolution equations of hyperbolic type. J. Fac. Sci., Univ. Tokyo, Sect. I 17, 241–258 (1970)

    MathSciNet  MATH  Google Scholar 

  21. Kozak, M.: A fundamental solution of a second-order differential equation in a Banach space. Univ. Iagellonicae Acta Math. 32, 275–289 (1995)

    MathSciNet  Google Scholar 

  22. Lebedev, L.P., Cloud, M.J.: Introduction to Mathematical Elasticity. World Scientific, Singapore (2009)

    Book  MATH  Google Scholar 

  23. Lin, Y.: Time-dependent perturbation theory for abstract evolution equations of second order. Stud. Math. 130, 263–274 (1998)

    MATH  Google Scholar 

  24. Lutz, D.: On bounded time-dependent perturbations of operator cosine functions. Aequ. Math. 23, 197–203 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lutz, D.: Periodische operatorwertige cosinusfunktionen. Results Math. 4, 75–83 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  26. Medeiros, L.A.: The initial value problem for nonlinear wave equation in the Hilbert space. Trans. Am. Math. Soc. 136, 305–327 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  27. Obrecht, E.: Evolution operators for higher order abstract parabolic equations. Czechoslov. Math. J. 36, 210–222 (1986)

    MathSciNet  Google Scholar 

  28. Obrecht, E.: The Cauchy problem for time-dependent abstract parabolic equations of higher order. J. Math. Anal. Appl. 125, 508–530 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  29. Peng, Y., Xiang, X.: Second-order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls. J. Ind. Manag. Optim. 4, 17–32 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Peng, Y., Xiang, X., Wei, W.: Second-order nonlinear impulsive integro-differential equations of mixed type with time-varying generating operators and optimal controls on Banach spaces. Comput. Math. Appl. 57, 42–53 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Radyno, Ya.V.: Differential equations in a Banach-space scale. Differ. Equ. 21, 971–979 (1985)

    MATH  Google Scholar 

  32. Radyno, Ya.V.: Vectors of exponential type in operational calculus and differential equations. Differ. Equ. 21, 1062–1070 (1985)

    MATH  Google Scholar 

  33. Serizawa, H., Watanabe, M.: Time-dependent perturbation for cosine families in Banach spaces. Houst. J. Math. 12, 579–586 (1986)

    MathSciNet  MATH  Google Scholar 

  34. Travis, C.C., Webb, G.F.: Compactness, regularity, and uniform continuity properties of strongly continuous cosine families. Houst. J. Math. 3, 555–567 (1977)

    MathSciNet  MATH  Google Scholar 

  35. Travis, C.C., Webb, G.F.: Second order differential equations in Banach space. In: Proc. Internat. Sympos. on Nonlinear Equations in Abstract Spaces, pp. 331–361. Academic Press, New York (1987)

    Google Scholar 

  36. Winiarska, T.: Evolution equations of second order with operator dependent on t. Sel. Probl. Math. Cracow Univ. Tech. 6, 299–314 (1995)

    MathSciNet  Google Scholar 

  37. Winiarska, T.: Quasilinear evolution equations with operators dependent on t. Mat. Stud. 21, 170–178 (2004)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The author wishes to thank the referee for their comments and suggestions.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidad de Santiago, USACH, Casilla 307, Correo 2, Santiago, Chile

    Hernán R. Henríquez

Authors
  1. Hernán R. Henríquez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hernán R. Henríquez.

Additional information

Communicated by Jerome A. Goldstein.

This research was supported in part by CONICYT under Grant FONDECYT 1130144 and DICYT-USACH.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Henríquez, H.R. Existence of solutions of the nonautonomous abstract Cauchy problem of second order. Semigroup Forum 87, 277–297 (2013). https://doi.org/10.1007/s00233-013-9485-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-013-9485-8

Keywords

Search

Navigation