Annals of Functional Analysis

Decay bounds for nonlocal evolution equations in Orlicz spaces

Uriel Kaufmann, Julio D. Rossi, and Raul Vidal

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We show decay bounds of the form

Rdϕ(u(x,t))dxCtμ for integrable and bounded solutions to the nonlocal evolution equation

ut(x,t)=RdJ(x,y)G(u(y,t)u(x,t))(u(y,t)u(x,t))dy+f(u(x,t)). Here G is a nonnegative and even function, and f verifies f(ξ)ξ0 for all ξ0. We remark that G is not assumed to be homogeneous. The function ϕ and the exponent μ depend on G via adequate hypotheses, while J is a nonnegative kernel satisfying suitable assumptions.

Article information

Source
Ann. Funct. Anal., Volume 7, Number 2 (2016), 261-269.

Dates
Received: 13 March 2015
Accepted: 9 July 2015
First available in Project Euclid: 29 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.afa/1456754403

Digital Object Identifier
doi:10.1215/20088752-3475634

Mathematical Reviews number (MathSciNet)
MR3465028

Zentralblatt MATH identifier
1337.47110

Subjects
Primary: 47G10: Integral operators [See also 45P05]
Secondary: 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25] 45G10: Other nonlinear integral equations

Keywords
Orlicz space nonlocal diffusion energy methods

Citation

Kaufmann, Uriel; Rossi, Julio D.; Vidal, Raul. Decay bounds for nonlocal evolution equations in Orlicz spaces. Ann. Funct. Anal. 7 (2016), no. 2, 261--269. doi:10.1215/20088752-3475634. https://projecteuclid.org/euclid.afa/1456754403


Export citation

References

  • [1] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi, and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Math. Surveys Monogr. 165, Amer. Math. Soc., Providence, 2010.
  • [2] P. Bates, X. Chen, and A. Chmaj, Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions, Calc. Var. Partial Differential Equations 24 (2005), no. 3, 261–281.
  • [3] E. Chasseigne, M. Chaves, and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. (9) 86 (2006), no. 3, 271–291.
  • [4] E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573.
  • [5] S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, to appear in Rev. Mat. Iberoam., preprint, arXiv:1407.3313 [math.AP].
  • [6] Q. Du, M. Gunzburger, R. Lehoucq, and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci. 23 (2013), no. 3, 493–540.
  • [7] P. Fife, “Some nonclassical trends in parabolic and parabolic-like evolutions” in Trends in Nonlinear Analysis, Springer, Berlin, 2003, 153–191.
  • [8] E. Harboure, O. Salinas, and B. Viviani, Relations between weighted Orlicz and $\mathrm{BMO}_{\phi}$ spaces through fractional integrals, Comment. Math. Univ. Carolin. 40 (1999), no. 1, 53–69.
  • [9] V. Hutson, S. Martínez, K. Mischaikow, and G. T. Vickers, The evolution of dispersal, J. Math. Biol. 47 (2003), no. 6, 483–517.
  • [10] L. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, J. Math. Pures Appl. (9) 92 (2009), no. 2, 163–187.
  • [11] R. O’Neil, Fractional integration in Orlicz spaces, I, Trans Amer. Math. Soc. 115 (1965), 300–328.
  • [12] M. L. Parks, R. Lehoucq, S. Plimpton, and S. Silling, Implementing peridynamics within a molecular dynamics code, Comput. Phys. Commun. 179 (2008), no. 11, 777–783.
  • [13] S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids 48 (2000), no. 1, 175–209.