Skip to main content
SpringerLink
Log in

Existence and uniqueness of solution for some time fractional parabolic equations involving the 1-Laplace operator

  • Original Paper
  • Published:
Partial Differential Equations and Applications Aims and scope Submit manuscript

Abstract

This paper is devoted to study the time fractional parabolic 1-Laplacian type. Firstly, by using the parabolic regularization technique and approximating the parabolic 1-Laplacian problem by a class of parabolic equations of p-Laplacian type with \(p>1\), we establish the existence of global weak radial solutions to the considered problem for a wide class of nonlinearities. Secondly, we discuss the extinction property of solutions to the time fractional total variation flow (FTVF) with different boundary conditions (Dirichlet and Neumann conditions). We conclude this paper by providing an example of explicit solution to the (FTVF).

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

Data availability statement

All data generated or analysed during this study are included in this published article (and its supplementary information files).

References

  1. Akagi, G.: Fractional flows driven by subdifferentials in Hilbert spaces. Isr. J. Math. 234(2), 809–862 (2019)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alves, C.O., Boudjeriou, T.: Existence of solution for a class of heat equation involving the \(1\)-Laplacian operator (Submitted)

  3. Alves, C.O.: A Berestycki-Lions type result for a class of problems involving the 1-Laplacian operator. Commun. Contemp. Math. (2021). https://doi.org/10.1142/S021919972150022X

    Article  Google Scholar 

  4. Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. oxford mathematical monographs. The Clarendon Press, Oxford University Press, New York (2000) (ISBN: 0-19-850245-1)

  5. Andreu, F., Ballester, C., Caselles, V., Mazón, J.M.: The Dirichlet problem for the total variation flow. J. Funct. Anal. 180, 347–403 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Andreu, F., Caselles, V., Díaz, J.I., Mazón, J.M.: Some qualitative properties for the total variation flow. J. Funct. Anal. 188, 516–547 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Andreu, F., Ballester, C., Caselles, V., Mazón, J.M.: Minimizing total variation flow. Differ. Integral Equ. 14, 321–360 (2001)

    MATH  MathSciNet  Google Scholar 

  8. Anzellotti, G.: Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. 135(1), 293–318 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  9. Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. MPS-SIAM, Philadelphia (2006)

    Book  MATH  Google Scholar 

  10. Bonforte, M., Figalli, A.: Total variation flow and sign fast diffusion in one dimension. J. Differ. Equ. 252, 4455–4480 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Clément, P.: On abstract Volterra equations in Banach spaces with completely positive kernels. Infnite-dimensional systems Retzhof,: Lecture Notes in Math., vol. 1076, pp. 3–40. Springer, Berlin (1983)

  12. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)

    MATH  Google Scholar 

  13. Giga, Y., Kohn, R.: Scale-invariant extinction time estimates for some singular diffusion equations. Discrete Contin. Dyn. Syst. 30(2), 509–535 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gianazz, U., Klaus, C.: p-parabolic approximation of total variation flow solutions. Indiana Univ. Math. J. 60(5), 1519–1550 (2019)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gripenberg, G.: Volterra integro-differential equations with accretive nonlinearity. J. Differ. Equ. 60, 57–79 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  16. Juutinen, P.: \(p\)-harmonic approximation of functions of least gradient. Indiana Univ. Math. J. 54, 1015–1029 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kawohl, B., Schuricht, F.: Dirichlet problems for the \(1-\)Laplace operator, including the eigenvalue problem. Commun. Contemp. Math. 9(4), 525–543 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kavian, O.: Introduction à la Théorie des Points Critiques et Applications aux Problèmes Elliptiques. Springer, Heidelberg (1993)

    MATH  Google Scholar 

  19. Prüss, J.: Evolutionary Integral Equations and Applications, Monographs in Mathematics, vol. 87. Birkhäuser Verlag, Basel (1993)

    Book  MATH  Google Scholar 

  20. Robinson, J.C.: Infinite-dimensional Dynamical Systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  21. Salas, A.M., de León, S.S.: Elliptic equations involving the 1-Laplacian and a subcritical source term. Nonlinear Anal. 168, 50–66 (2018)

    Article  MATH  MathSciNet  Google Scholar 

  22. Samko, S., Cardoso, R.P.: Integral equations of the first kind of Sonine type. Int. J. Math. Math. Sci. 57, 3609–3632 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Segura de León, S., Webler, C.: Global existence and uniqueness for the inhomogeneous 1-Laplace evolution equation. NoDEA Nonlinear Differ. Equ. Appl. 22, 1213–1246 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  24. Simon, J.: Compact sets in the space \(L^{p}(0, T, B)\). Ann. Math. Pura Appl. 146(4), 65–96 (1987)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Vergara, V., Zacher, R.: Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods. SIAM J. Math. Anal. 47, 210–239 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  27. Vergara, V., Zacher, R.: A priori bounds for degenerate and singular evolutionary partial integro-differential equations. Nonlinear Anal. 73, 3572–3585 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  28. Vergara, V., Zacher, R.: Stability, instability, and blowup for time fractional and other nonlocal in time semilinear subdiffusion equations. J. Evol. Equ. 17, 599–626 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  29. Wittbold, P., Wolejko, P., Zacher, R.: Bounded weak solutions of time-fractional porous medium type and more general nonlinear and degenerate evolutionary integro-differential equations. JMAA 499(1), 125007 (2021)

    MATH  MathSciNet  Google Scholar 

  30. Zacher, R.: Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces. Funkcial. Ekvac. 52, 1–18 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  31. Zacher, R.: Maximal regularity of type \(L_{p}\) for abstract parabolic Volterra equations. J. Evol. Equ. 5, 79–103 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. Zacher, R.: Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients. J. Math. Anal. Appl. 348, 137–149 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  33. Zacher, R.: Time fractional diffusion equations: solution concepts, regularity, and long-time behavior. In: Handbook of Fractional Calculus with Applications, vol. 2, pp. 159–179. De Gruyter, Berlin (2019)

  34. Zheng, S.: Nonlinear Evolution Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 133. Chapman & Hall/CRC, Boca Raton (2004)

    Google Scholar 

  35. Ziemer, W.P.: Weakly Differentiable Functions, GTM 120. Springer, Berlin (1989)

    Book  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the reviewers for their time to read the manuscript and kind comments and suggestions. C. O. Alves was supported by CNPq/Brazil 307045/2021-8 and Projeto Universal FAPESQ-PB 3031/2021.

Author information

Authors and Affiliations

  1. Unidade Acadêmica de Matemática Universidade Federal de Campina Grande, Campina Grande, PB, 58429-970, Brazil

    Claudianor O. Alves

  2. Department of Basic Teaching, Institute of Electrical and Electronic Engineering University of Boumerdes, 35000, Boumerdes, Algeria

    Tahir Boudjeriou

Authors
  1. Claudianor O. Alves
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tahir Boudjeriou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tahir Boudjeriou.

Ethics declarations

Conflict of interest

There are no undeclared competing interests or funding.

Additional information

This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.

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

Alves, C.O., Boudjeriou, T. Existence and uniqueness of solution for some time fractional parabolic equations involving the 1-Laplace operator. Partial Differ. Equ. Appl. 4, 5 (2023). https://doi.org/10.1007/s42985-022-00222-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s42985-022-00222-y

Keywords

Mathematics Subject Classification

Search

Navigation