Skip to main content

Asymptotic behavior and representation of solutions to a Volterra kind of equation with a singular kernel

Abstract

Let A be a densely defined closed, linear \(\omega\)-sectorial operator of angle \(\theta \in [0,\frac{\pi }{2})\) on a Banach space X, for some \(\omega \in \mathbb {R}\). We give an explicit representation (in terms of some special functions) and study the precise asymptotic behavior as time goes to infinity of solutions to the following diffusion equation with memory: \(\displaystyle u'(t)=Au(t)+(\kappa *Au)(t), \, t >0\), \(u(0)=u_0\), associated with the (possible) singular kernel \(\kappa (t)=\alpha e^{-\beta t}\frac{t^{\mu -1}}{\Gamma (\mu )},\;\;t>0\), where \(\alpha \in \mathbb {R}\), \(\alpha \ne 0\), \(\beta \ge 0\) and \(0<\mu < 1\).

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Chen, J.-H., Liang, J., Xiao, T.-J.: Stability of solutions to integro-differential equations in Hilbert spaces. Bull. Belg. Math. Soc. Simon Stevin 18(5, [Supplement 2010 on cover]), 781–792 (2011)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Chen, J.-H., Xiao, T.-J., Liang, J.: Uniform exponential stability of solutions to abstract Volterra equations. J. Evol. Equ. 9(4), 661–674 (2009)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Coleman, B.D., Gurtin, M.E.: Equipresence and constitutive equations for rigid heat conductors. Z. Angew. Math. Phys. 18, 199–208 (1967)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Corduneanu, C.: Integral Equations and Applications. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

  5. 5.

    Cuesta, E.: Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. In: Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, pp. 277–285 (2007)

  6. 6.

    Engel, K.-J., Nagel, R.: One-parameter semigroups for linear evolution equations. In: Graduate Texts in Mathematics (With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt), vol. 194. Springer, New York (2000)

  7. 7.

    Gal, C.G., Warma, M.: Fractional in Time Semilinear Parabolic Equations and Applications, Mathématiques and Applications Series, vol. 84. Springer, Berlin (2020)

    MATH  Google Scholar 

  8. 8.

    Gripenberg, G., Londen, S.-O., Staffans, O.: Volterra Integral and Functional Equations, Encyclopedia of Mathematics and Its Applications, vol. 34. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  9. 9.

    Haak, B.H., Jacob, B., Partington, J.R., Pott, S.: Admissibility and controllability of diagonal Volterra equations with scalar inputs. J. Differ. Equ. 246(11), 4423–4440 (2009)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Haase, M.: The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, vol. 169. Birkhäuser Verlag, Basel (2006)

    Book  Google Scholar 

  11. 11.

    Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag-Leffler functions and their applications. J. Appl. Math., Art. ID 298628, 51 (2011)

  12. 12.

    Jacob, B., Partington, J.R.: A resolvent test for admissibility of Volterra observation operators. J. Math. Anal. Appl. 332(1), 346–355 (2007)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Keyantuo, V., Lizama, C., Warma, M.: Asymptotic behavior of fractional-order semilinear evolution equations. Differ. Integr. Equ. 26(7–8), 757–780 (2013)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Lizama, C.: Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243(2), 278–292 (2000)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Lizama, C., Ponce, R.: Bounded solutions to a class of semilinear integro-differential equations in Banach spaces. Nonlinear Anal. 74(10), 3397–3406 (2011)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Lizama, C., Sánchez, J.: On perturbation of \(K\)-regularized resolvent families. Taiwan. J. Math. 7(2), 217–227 (2003)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Mainardi, F., Gorenflo, R.: On Mittag–Leffler-type functions in fractional evolution processes. In: Higher Transcendental Functions and Their Applications, vol. 118, pp. 283–299 (2000)

  18. 18.

    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  19. 19.

    Nunziato, J.W.: On heat conduction in materials with memory. Quart. Appl. Math. 29, 187–204 (1971)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Prüss, J.: Evolutionary integral equations and applications. In: Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel ([2012] reprint of the 1993 edition) (1993)

  21. 21.

    Vergara, V., Zacher, R.: Lyapunov functions and convergence to steady state for differential equations of fractional order. Math. Z. 259(2), 287–309 (2008)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

This research has been partially supported by Conicyt, Project MEC #80180010. The work of M. Warma is partially supported by Air Force Office of Scientific Research (AFOSR) under Award NO [FA9550-18-1-0242] and Army Research Office (ARO) under Award NO: W911NF-20-1-0115.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Rodrigo Ponce.

Additional information

Publisher's Note

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

Communicated by Abdelaziz Rhandi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ponce, R., Warma, M. Asymptotic behavior and representation of solutions to a Volterra kind of equation with a singular kernel. Semigroup Forum 102, 250–273 (2021). https://doi.org/10.1007/s00233-020-10157-8

Download citation

Keywords

  • Volterra kind of equation
  • Generalized Mittag-Leffler functions
  • Explicit representation of solutions
  • Exponential stability of solutions
  • Heat equation with memory