## Annals of Functional Analysis

### Decay bounds for nonlocal evolution equations in Orlicz spaces

#### Abstract

We show decay bounds of the form

$$\int_{\mathbb{R}^{d}}\phi (u(x,t))\,dx\leqCt^{-\mu}$$ for integrable and bounded solutions to the nonlocal evolution equation

$$u_{t}(x,t)=\int_{\mathbb{R}^{d}}J(x,y)G(u(y,t)-u(x,t))(u(y,t)-u(x,t))\,dy+f(u(x,t)).$$ Here $G$ is a nonnegative and even function, and $f$ verifies $f(\xi)\xi\leq0$ for all $\xi\geq0$. We remark that $G$ is not assumed to be homogeneous. The function $\phi$ and the exponent $\mu$ 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
Accepted: 9 July 2015
First available in Project Euclid: 29 February 2016

https://projecteuclid.org/euclid.afa/1456754403

Digital Object Identifier
doi:10.1215/20088752-3475634

Mathematical Reviews number (MathSciNet)
MR3465028

Zentralblatt MATH identifier
1337.47110

#### 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

#### 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.