Skip to main content
Log in

On a fractional reaction–diffusion equation

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This paper is devoted to study the global well-posedness and spatiotemporal asymptotic behavior of solutions for a fractional reaction–diffusion equation.

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

Access this article

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. Avdonin, S., Pandolfi, L.: Simultaneous temperature and flux controllability for heat equations with memory. Q. Appl. Math. 71(2), 339–368 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bae, H.-O., Jin, B.J.: Temporal and spatial decays for the Navier–Stokes equations. Proc. R. Soc. Edinburgh Sect. A 135(3), 461–477 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carillo, S.: Existence, uniqueness and exponential decay: an evolution problem in heat conduction with memory. Q. Appl. Math. 69(4), 635–649 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carillo, S.: Some remarks on materials with memory: heat conduction and viscoelasticity. J. Nonlinear Math. Phys. 12(suppl. 1), 163–178 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carillo, S., Valente, V., Vergara Caffarelli, G.: A linear viscoelasticity problem with a singular memory kernel: an existence and uniqueness result. Differ. Integral Equ. 26(9–10), 1115–1125 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Carillo, S., Valente, V., Vergara Caffarelli, G.: Heat conduction with memory: a singular kernel problem. Evol. Equ. Control Theory 3(3), 399–410 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. de Almeida, M.F., Viana, A.: Self-similar solutions for a superdiffusive heat equation with gradient nonlinearity. Electron. J. Differ. Equ. 2016(250), 1–20 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Desch, W., Grimmer, R.: Singular relaxation moduli and smoothing in three-dimensional viscoelasticity. Trans. Am. Math. Soc. 314(1), 381–404 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Desch, W., Grimmer, R.: Smoothing properties of linear Volterra integro-differential equations. SIAM J. Math. Anal. 20(1), 116–132 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deseri, L., Zingales, M., Pollaci, P.: The state of fractional hereditary materials (FHM). Discret. Contin. Dyn. Syst. Ser. B 19(7), 2065–2089 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fabrizio, M.: Fractional rheological models for thermomechanical systems. Dissipation and free energies, Fract. Calc. Appl. Anal. 17(1), 206–223 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Ferreira, L.C.F., Villamizar-Roa, E.J.: Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations. Differ. Integral Equ. 19(12), 1349–1370 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Gentili, G.: Regularity and stability for a viscoelastic material with a singular memory kernel. J. Elast. 37(2), 139–156 (1994/95)

  14. Giga, Yoshikazu: Solutions for semilinear parabolic equations in \(L^{p}\) and regularity of weak solutions of the Navier-Stokes system. J. Differ. Equ. 62(2), 186–212 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. Giorgi, C., Pata, V.: Asymptotic behavior of a nonlinear hyperbolic heat equation with memory. NoDEA Nonlinear Differ. Equ. Appl. 8(2), 157–171 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gurtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31(2), 113–126 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hanyga, A.: Fractional-order relaxation laws in non-linear viscoelasticity. Contin. Mech. Thermodyn. 19(1–2), 25–36 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Koeller, R.C.: Applications of fractional calculus to the theory of viscoelasticity. Trans. ASME J. Appl. Mech. 51(2), 299–307 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kukavica, I., Torres, J.J.: Weighted \(L^{p}\) decay for solutions of the Navier–Stokes equations. Commun. Partial Differ. Equ. 32(4–6), 819–831 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imp. Coll. Press, London (2010)

    Book  MATH  Google Scholar 

  21. Miyakawa, T.: Notes on space-time decay properties of nonstationary incompressible Navier–Stokes flows in \({ R}^{n}\). Funkcial. Ekvac. 45(2), 271–289 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Miyakawa, T.: On space-time decay properties of nonstationary incompressible Navier–Stokes flows in \({ R}^{n}\). Funkcial. Ekvac. 43(3), 541–557 (2000)

    MathSciNet  MATH  Google Scholar 

  23. Pandolfi, L.: The controllability of the Gurtin-Pipkin equation: a cosine operator approach. Appl. Math. Optim. 52(2), 143–165 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Prüss, J.: Evolutionary integral equations and applications, Monographs in Mathematics, 87. Birkhäuser, Basel (1993)

    Book  Google Scholar 

  25. Schneider, W.R., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30(1), 134–144 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tatar, N.: Exponential decay for a viscoelastic problem with a singular kernel. Z. Angew. Math. Phys. 60(4), 640–650 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang, R.-N., Chen, D.-H., Xiao, T.-J.: Abstract fractional Cauchy problems with almost sectorial operators. J. Differential Equations 252(1), 202–235 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, J.-M., Guo, B.-Z., Fu, M.-Y.: Dynamic behavior of a heat equation with memory. Math. Methods Appl. Sci. 32(10), 1287–1310 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Weissler, F.B.: Existence and non-existence of global solutions for a semilinear heat equation. Israel J. Math. 38, 29–40 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wiegner, M.: Decay and stability in \(L_p\) for strong solutions of the Cauchy problem for the Navier-Stokes equations, In: The Navier-Stokes equations (Oberwolfach, 1988), 95–99, Lecture Notes in Math., 1431, Springer, Berlin

  31. Yong, J., Zhang, X.: Heat equation with memory in anisotropic and non-homogeneous media. Acta Math. Sin. (Engl. Ser.) 27(2), 219–254 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Arlúcio Viana.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Andrade, B., Viana, A. On a fractional reaction–diffusion equation. Z. Angew. Math. Phys. 68, 59 (2017). https://doi.org/10.1007/s00033-017-0801-0

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-017-0801-0

Mathematics Subject Classification

Keywords

Navigation