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Non-local Diffusion Equations Involving the Fractional \(p(\cdot )\)-Laplacian

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Abstract

In this paper we study a class of nonlinear quasi-linear diffusion equations involving the fractional \(p(\cdot )\)-Laplacian with variable exponents, which is a fractional version of the nonhomogeneous \(p(\cdot )\)-Laplace operator. The paper is divided into two parts. In the first part, under suitable conditions on the nonlinearity f, we analyze the problem \(({\mathscr {P}}_{1})\) in a bounded domain \(\varOmega \) of \({\mathbb {R}}^N\) and we establish the well-posedness of solutions by using techniques of monotone operators. We also study the large-time behaviour and extinction of solutions and we prove that the fractional \(p(\cdot )\)-Laplacian operator generates a (nonlinear) submarkovian semigroup on \(L^{2}(\varOmega ).\) In the second part of the paper we establish the existence of global attractors for problem \(({\mathscr {P}}_{2})\) under certain conditions in the potential \({\mathbb {V}}.\) Our results are new in the literature, both for the case of variable exponents and for the fractional p-laplacian case with constant exponent.

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References

  1. Akagi, G., Matsuura, K.: Nonlinear diffusion equations driven by the \(p(\cdot )\)-Laplacian. NoDEA Nonlinear Differ. Equ. Appl. 20(1), 37–64 (2013)

    Article  MathSciNet  Google Scholar 

  2. Bahrouni, A.: On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete Contin. Dyn. Syst. Ser. S 11(3), 379–389 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Barbu, L., Moroşanu, G.: Elliptic-like regularization of a fully nonlinear evolution inclusion and applications. Commun. Contemp. Math. 19(5), 1650037 (2017)

    Article  MathSciNet  Google Scholar 

  4. Barvu, V.: Nonlinear Semigroups and Differential Equations in Banach Space. Noordhoff International, Leyden (1976)

    Google Scholar 

  5. Brezis, H.: Operateurs maximaus monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Company, Amsterdam (1973)

    MATH  Google Scholar 

  6. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitex. Springer, New York (2011)

    MATH  Google Scholar 

  7. Brasco, L., Parini, E., Squassina, M.: Stability of variational eigenvalues for the fractional p-Laplacian. Discrete Contin. Dyn. Syst. 36, 1813–1845 (2016)

    Article  MathSciNet  Google Scholar 

  8. Bogdan, K., Burdzy, K., Chen, Z.-Q.: Censored stable processes. Probab. Theory Relat. Fields 127(1), 89–152 (2003)

    Article  MathSciNet  Google Scholar 

  9. Caffarelli, L.: Nonlocal Equations, Drifts and Games. Nonlinear Partial Differential Equations, pp. 37–52. Springer, Berlin (2012)

    Book  Google Scholar 

  10. Cipriani, F., Grillo, G.: Nonlinear Markov semigroups, nonlinear Dirichlet forms and applications to minimal surfaces. J. Reine Angew. Math. 562, 201–235 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Cholewa, J.W., Dlotko, T.: Global Attractors in Abstract Parabolic Problems. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  12. Del Pezzo, L.M., Rossi, J.D.: Traces for fractional Sobolev spaces with variable exponents. Adv. Oper. Theory 2(4), 435–446 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Diening, L., Harjulehto, P., Hasto, P., Ruzicka, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)

    Book  Google Scholar 

  14. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  15. Fan, X.: Boundary trace embedding theorems for variable exponent Sobolev spaces. J. Math. Anal. Appl. 339, 1395–1412 (2008)

    Article  MathSciNet  Google Scholar 

  16. Fan, X.: Solutions for \(p(x)\)-Laplacian Dirichlet problems with singular coefficients. J. Math. Anal. Appl. 312, 464–477 (2005)

    Article  MathSciNet  Google Scholar 

  17. Fu, Y., Pucci, P.: On solutions of space-fractional diffusion equations by means of potential wells. Electron. J. Qual. Theory Differ. Equ. 70, 1–17 (2016)

    Article  MathSciNet  Google Scholar 

  18. Franzina, G., Palatucci, G.: Fractional p-eigenvalues. Riv. Math. Univ. Parma (N.S.) 5, 373–386 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Giacomoni, J., Tiwari, S.: Existence and global behavior of solutions to fractional \(p\)-Laplacian parabolic problems. Electron. J. Differ. Equ. 44, 1–20 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Gilboa, P., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)

    Article  MathSciNet  Google Scholar 

  21. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Mathematical Studies Monograph, vol. 24. Pitman, Boston (1985)

    MATH  Google Scholar 

  22. Iannizzotto, A., Shibo, S.L., Perera, K., Squassina, M.: Existence results for fractional p-Laplacian problems via Morse theory. Adv. Calc. Var. 9, 101–125 (2016)

    Article  MathSciNet  Google Scholar 

  23. Iannizzotto, A., Mosconi, S., Squassina, M.: Global Hölder regularity for the fractional p-Laplacian. Preprint, arXiv:1411.2956 (2018)

  24. Kaufmann, U., Rossi, J.D., Vidal, R.: Fractional Sobolev spaces with variable exponents and fractional \(p(x)\)-Laplacians. Electron J. Qual. Theory Differ. Equ. 76, 1–10 (2017)

    Article  MathSciNet  Google Scholar 

  25. Kovăcik, O., Răkosnik, J.: On spaces \(L^{p(x)}\) and \(W^{k,p(x)}\). Czechoslovak Math. J. 41, 592–618 (1991)

    MathSciNet  MATH  Google Scholar 

  26. Laskin, N.: Fractional quantum mechanics and Levy path integrals. Phys. Lett. A 268(4–6), 298–305 (2000)

    Article  MathSciNet  Google Scholar 

  27. Lindgren, E., Lindqvist, P.: Fractional eigenvalues. Calc. Var. Partial Differ. Equ. 49, 795–826 (2014)

    Article  MathSciNet  Google Scholar 

  28. Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)

    Article  MathSciNet  Google Scholar 

  29. Mellet, A., Mischler, S., Mouhot, C.: Fractional diffusion limit for collisional kinetic equations. Arch. Ration. Mech. Anal. 199, 493–525 (2011)

    Article  MathSciNet  Google Scholar 

  30. Simon, J.: Régularité de la solution d’une équation non lineéaire dans \({\mathbb{R}}^{N}\). In: Benilan, P., Robert, J. (eds.) Journés d’Analyse Non Linéaire. Lecture Notes in Mathematics, vol. 655, pp. 205–227. Springer, Berlin (1978)

    Chapter  Google Scholar 

  31. Vázquez, J.L.: The Dirichlet problem for the fractional \(p\)-Laplacian evolution equation. J. Differ. Equ. 260(7), 6038–6056 (2016)

    Article  MathSciNet  Google Scholar 

  32. Velez-Santiago, A., Warma, M.: A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions. J. Math. Anal. Appl. 372, 120–139 (2010)

    Article  MathSciNet  Google Scholar 

  33. Willem, M.: Minimax Theorems. Birkhauser, Boston (1996)

    Book  Google Scholar 

  34. Warma, M.: Local Lipschitz continuity of the inverse of the fractional \(p\)-Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains. Nonlinear Anal. 135, 129–157 (2016)

    Article  MathSciNet  Google Scholar 

  35. Warma, M.: The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets. Potential Anal. 42, 499–547 (2015)

    Article  MathSciNet  Google Scholar 

  36. Zeidler, E.: Nonlinear Functional Analysis and Its Applications. Springer, Berlin (1990)

    Book  Google Scholar 

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Acknowledgements

The author would like to thanks Professor Juan Rocha Barriga and Professor Lauren Maria Mezzomo Bonaldo for their suggestions and fruitful discussions. The author would like to thank the anonymous referee by the careful reading and all of his/her valuable comments.

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  1. Departamento de Matemática e Computação, Faculdade de Ciências e Tecnologia, UNESP - Universidade Estadual Paulista, Presidente Prudente, SP, 19060-900, Brazil

    Elard J. Hurtado

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Hurtado, E.J. Non-local Diffusion Equations Involving the Fractional \(p(\cdot )\)-Laplacian. J Dyn Diff Equat 32, 557–587 (2020). https://doi.org/10.1007/s10884-019-09745-2

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