Skip to main content
SpringerLink
Log in

A Note on the Spectral Analysis of Some Fourth-Order Differential Equations with a Semigroup Approach

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we analyze the solubility of a class of abstract fourth-order in time linear evolution equations, using the roots of the characteristic polynomial that is associated with the equations.

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. Abouelregal, A.E., Sedighi, H.M., Eremeyev, V.A.: Thermomagnetic behavior of a semiconductor material heated by pulsed excitation based on the fourth-order MGT photothermal model. Continuum Mech. Thermodyn. (2022). https://doi.org/10.1007/s00161-022-01170-z

    Article  MATH  Google Scholar 

  2. Amann, H.: Linear and quasilinear parabolic problems. Volume I: Abstract Linear Theory, Birkhäuser Verlag, Basel (1995)

  3. Bezerra, F.D.M., Santos, L.A.: Fractional powers approach of operators for abstract evolution equations of third order in time. J. Differ. Equ. 269(7), 5661–5679 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bezerra, F.D.M., Santos, L.A.: Chebyshev polynomials for higher order differential equations and fractional powers. Math. Ann. (2022). https://doi.org/10.1007/s00208-022-02554-x

    Article  Google Scholar 

  5. Bezerra, F. D. M., Santos, L. A.: An extended version of the Cayley–Hamilton–Ziebur Theorem (2022). arxiv:2210.01976

  6. Caixeta, A.H., Lasiecka, I., Cavalcanti, V.N.D.: Global attractors for a third order in time nonlinear dynamics. J. Differ. Equ. 261(1), 113–147 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dell’Oro, F., Pata, V.: On a fourth-order equation of Moore–Gibson–Thompson type. Milan J. Math. 85, 215–234 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fattorini, H.O.: The cauchy problem. In: Encyclopedia of Mathematics and its Applications, vol. 18. Addison-Wesley Publishing Company, Reading, Massachusetts (1983)

    Google Scholar 

  9. Fattorini, H.O.: Ordinary differential equations in linear topological spaces, I. J. Differ. Equ. 5, 72–105 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hasse, M.: The Functional Calculus for Sectorial Operators. Birkhäuser Verlag (2006)

  11. Henry, D.: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, vol. 840. Springer-Verlag, Berlin (1981)

  12. Kaltenbacher, B., Lasiecka, I., Marchand, R.: Well-posedness and exponential decay rates for the Moore–Gibson–Thompson equation arising in high intensity ultrasound. Control Cybernet. 40(4), 971–988 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Kreǐn, S. G.: Linear differential equations in a banach space. Transl. Mathem. Monogr., 29, American Mathematical Soc. (1971)

  14. Lange, S.: Algebra, Graduate Texts in Mathematics 211, 3rd edn. Springer-Verlag, New York (2002)

    Google Scholar 

  15. Liu, W., Chen, Z., Tiu, Z.: New general decay result for a fourth-order Moore–Gibson–Thompson equation with memory. Electron. Res. Arch. 28(1), 433–457 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lizama, C., Murillo, M.: Well-posedness for a fourth-order equation of Moore–Gibson–Thompson type. Electron. J. Qual. Theory Differ. Equ. 81, 1–18 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Marchand, R., McDevitt, T., Triggiani, R.: An abstract semigroup approach to the third-order Moore–Gibson–Thompson partial differential equation arising in high-intensity ultrasound: structural decomposition, spectral analysis, exponential stability. Math. Methods Appl. Sci. 35(15), 1896–1929 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Martínez, C., Sanz, M.: Spectral mapping theorem for fractional powers in locally convex spaces, Ann. Scuola Norm. - SCI \(4^e\) série, tome 24(4), 685–702 (1997)

  19. Murillo-Arcila, M.: Well-posedness for the fourth-order Moore–Gibson–Thompson equation in the class of Banach-space-valued Hölder-continuous functions. Math. Methods Appl. Sci. 46(2), 1928–1937 (2023)

    Article  MathSciNet  Google Scholar 

  20. Mesloub, A., Zarai, A., Mesloub, F., Cherif, B.B., Abdalla, M.: The Galerkin method for fourth-order equation of the Moore–Gibson–Thompson type with integral condition. Adv. Math. Phys. (2021). https://doi.org/10.1155/2021/5532691

    Article  MathSciNet  Google Scholar 

  21. Mesloub, F., Merah, A., Boulaaras, S.: Solution blow-up for a fractional fourth-order equation of Moore–Gibson–Thompson type with nonlinearity nonlocal in time. Math. Notes 113(1), 72–80 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  22. Neubrander, F.: Well-posedness of higher order abstract Cauchy problems. Trans. Am. Math. Soc. 295, 257–290 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pazy, A.: Semigroup of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983)

  24. Pellicer. M., Said-Houari, B.: Wellposedness and decay rates for the cauchy problem of the Moore–Gibson–Thompson equation arising in high intensity ultrasound. Appl. Math. Optim. 1–32 (2017)

  25. Pellicer, M., Solà-Morales, J.: Optimal scalar products in the Moore–Gibson–Thompson equation. Evol. Equ. Control Theory 8(1), 203–220 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sobolevskiĭ, P.E.: Equations of parabolic type in a Banach space. Am. Math. Soc. Transl. 49, 1–62 (1966)

    Google Scholar 

  27. Triebel, H.: Interpolation theory, function spaces, differential operators, Veb Deutscher (1978)

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their comments and suggestions which greatly improved the work.

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal da Paraíba, João Pessoa, PB,  58051-900, Brazil

    Flank D. M. Bezerra & Maria J. M. Silva

  2. Instituto Federal da Paraíba, João Pessoa, PB,  58051-900, Brazil

    Lucas A. Santos

  3. Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador São-Carlense, 400, Centro, São Carlos, SP,  13566590, Brazil

    Carlos R. Takaessu Jr.

Authors
  1. Flank D. M. Bezerra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lucas A. Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Maria J. M. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Carlos R. Takaessu Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lucas A. Santos.

Ethics declarations

Conflict of interest

Data sharing does not apply to this article as no datasets were generated or analyzed during the current study. The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

F. D. M. Bezerra: Research partially supported by CNPq/Finance Code # 303039/2021-3, Brazil. C. R. Takaessu: Research partially supported by CAPES-PROEX-11169228/D and by FAPESP # 2020/14353-6, Brazil.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bezerra, F.D.M., Santos, L.A., Silva, M.J.M. et al. A Note on the Spectral Analysis of Some Fourth-Order Differential Equations with a Semigroup Approach. Results Math 78, 220 (2023). https://doi.org/10.1007/s00025-023-01999-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-023-01999-z

Keywords

Mathematics Subject Classification

Search

Navigation