# American Institute of Mathematical Sciences

March  2018, 38(3): 1365-1403. doi: 10.3934/dcds.2018056

## Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian

 1 Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile 2 BCAM -Basque Center for Applied Mathematics, Alameda de Mazarredo 14,48009 Bilbao, Spain

* Corresponding author: Carlos Lizama.

Received  May 2017 Published  December 2017

Fund Project: The first author is partially supported by FONDECYT grant number 1140258 and CONICYTPIA-Anillo ACT1416. The second author is partially supported by grant MTM2015-65888-C04-4-P from the Government of Spain, by the Basque Government through the BERC 2014-2017 program, by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and by a 2017 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation. The Foundation accepts no responsibility for the opinions, statements and contents included in the project and/or the results thereof, which are entirely the responsibility of the authors.

We study the equations
 \begin{align}\partial_t u(t, n) = L u(t, n) + f(u(t, n), n); \partial_t u(t, n) = iL u(t, n) + f(u(t, n), n)\end{align}
and
 \begin{align}\partial_{tt} u(t, n) =Lu(t, n) + f(u(t, n), n), \end{align}
where
 $n∈ \mathbb{Z}$
,
 $t∈ (0, ∞)$
, and
 $L$
is taken to be either the discrete Laplacian operator
 $Δ_\mathrm{d} f(n)=f(n+1)-2f(n)+f(n-1)$
, or its fractional powers
 $-(-Δ_{\mathrm{d}})^{σ}$
,
 $0<σ<1$
. We combine operator theory techniques with the properties of the Bessel functions to develop a theory of analytic semigroups and cosine operators generated by
 $Δ_\mathrm{d}$
and
 $-(-Δ_\mathrm{d})^{σ}$
. Such theory is then applied to prove existence and uniqueness of almost periodic solutions to the above-mentioned equations. Moreover, we show a new characterization of well-posedness on periodic Hölder spaces for linear heat equations involving discrete and fractional discrete Laplacians. The results obtained are applied to Nagumo and Fisher-KPP models with a discrete Laplacian. Further extensions to the multidimensional setting
 $\mathbb{Z}^N$
are also accomplished.
Citation: Carlos Lizama, Luz Roncal. Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1365-1403. doi: 10.3934/dcds.2018056
##### References:
 [1] L. Abadías, M. de León-Contreras and J. L. Torrea, Non-local fractional derivatives. Discrete and continuous, J. Math. Anal. Appl., 449 (2017), 734-755.  doi: 10.1016/j.jmaa.2016.12.006. [2] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables National Bureau of Standards Applied Mathematics Series, no. 55, U. S. Government Printing Office, Washington, DC, 1964. [3] E. C. Aifantis, Continuum Nanomechanics for Nanocrystalline and Ultrafine Grain Materials Materials Science and Engineering, 63, 6th International Conference on Nanomaterials by Severe Plastic Deformation, 2014. doi: 10.1088/1757-899X/63/1/012129. [4] W. Arendt, C. Batty and S. Bu, Fourier multipliers for Hölder continuous functions and maximal regularity, Studia Math., 160 (2004), 23-51.  doi: 10.4064/sm160-1-2. [5] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems. Second edition. Monographs in Mathematics, 96. Birkhäuser/Springer Basel AG, Basel, 2011. xii+539 pp. [6] H. Bateman, Some simple differential difference equations and the related functions, Bull. Amer. Math. Soc., 49 (1943), 494-512.  doi: 10.1090/S0002-9904-1943-07927-X. [7] J. J. Betancor, A. J. Castro, J. C. Fariña and L. Rodríguez-Mesa, Discrete harmonic analysis associated with ultraspherical expansions, preprint, arXiv:1512.01379. [8] S. Bochner, Curvature and Betti numbers in Real and complex vector bundles, Univ. e Politec. di Torino. Rend. Sem. Mat., 15 (1955/56), 225-253. [9] J. Campbell, The SMM model as a boundary value problem using the discrete diffusion equation, Theor. Population Biol., 72 (2007), 539-546.  doi: 10.1016/j.tpb.2007.08.001. [10] S-N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478. [11] Ó. Ciaurri, T. A. Gillespie, L. Roncal, J. L. Torrea and J. L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math., 132 (2017), 109-131.  doi: 10.1007/s11854-017-0015-6. [12] Ó. Ciaurri, C. Lizama, L. Roncal and J. L. Varona, On a connection between the discrete fractional Laplacian and superdiffusion, Appl. Math. Letters, 49 (2015), 119-125.  doi: 10.1016/j.aml.2015.05.007. [13] Ó. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea and J. L. Varona, Fractional discrete Laplacian versus discretized fractional Laplacian, preprint, arXiv:1507.04986. [14] Ó. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea and J. L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, preprint, arXiv:1608.08913. [15] C. Corduneanu, Almost Periodic Functions Interscience Tracts in Pure and Applied Mathematics, New York-London-Sydney, 1968. [16] O. Defterli, M. D'Elia, Q. Du, M. Gunzburger, R. Lehoucq and M. M. Meerschaert, Fractional diffusion on bounded domains, Fract. Calc. Appl. Anal., 18 (2015), 342-360. [17] R. E. Edwards, Fourier Series: A Modern Introduction, Vol. 2, Second ed. , Springer-Verlag, 1982. [18] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194, Springer, New York, 2000. [19] G. Fath, Propagation failure of traveling waves in a discrete bistable medium, Phys. D, 116 (1998), 176-190.  doi: 10.1016/S0167-2789(97)00251-0. [20] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces North-Holland Mathematics Studies, 108. Notas de Matemática, 99. North-Holland Publishing Co. , Amsterdam, 1985. [21] A. Favini, R. Labbas, S. Maingot, H. Tanabe and A. Yagi, Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces, Discrete Contin. Dyn. Syst., 22 (2008), 973-987.  doi: 10.3934/dcds.2008.22.973. [22] W. Feller, An Introduction to Probability Theory and Its Applications. Vol. 1. Third edition. John Wiley & Sons, Inc. , New York-London-Sydney, 1968. [23] W. Feller, An Introduction to Probability Theory and Its Applications. Vol. 2. Second edition. John Wiley & Sons, Inc. , New York-London-Sydney, 1971. [24] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Second Edition Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer-Verlag, Berlin, 1983. [25] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Seventh Edition Elsevier Academic Press, New York, 2007. [26] F. A. Grünbaum, "The bispectral problem: An overview", in Special functions 2000: current perspective and future directions (Tempe, AZ), 129-140, NATO Sci. Ser. Ⅱ Math. Phys. Chem. 30, Kluwer Acad. Publ., Dordrecht, 2001. [27] F. A. Grünbaum and P. Iliev, Heat kernel expansions on the integers, Math. Phys. Anal. Geom., 5 (2002), 183-200.  doi: 10.1023/A:1016258207606. [28] J.-S. Guo and C.-C. Wu, The existence of traveling wave solutions for a bistable three-component lattice dynamical system, J. Differential Equations, 260 (2016), 1445-1455.  doi: 10.1016/j.jde.2015.09.036. [29] C. Hu and B. Li, Spatial dynamics for lattice differential equations with a shifting habitat, J. Differential Equations, 259 (2015), 1967-1989.  doi: 10.1016/j.jde.2015.03.025. [30] I. Lasiecka and M. Wilke, Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system, Discrete Contin. Dyn. Syst., 33 (2013), 5189-5202.  doi: 10.3934/dcds.2013.33.5189. [31] N. N. Lebedev, Special Functions and Its Applications Dover, New York, 1972. [32] J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dynam. Differential Equations, 11 (1999), 49-127.  doi: 10.1023/A:1021841618074. [33] J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, Dynamical Systems, Lecture Notes in Math., Springer, Berlin, 1822 (2003), 231-298. [34] C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators North-Holland Mathematics Studies, vol. 187, North-Holland Publishing Co. , Amsterdam, 2001. [35] F. W. J. Olver and L. C. Maximon, Bessel Functions, NIST handbook of mathematical functions (edited by F. W. F. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark), Chapter 10, National Institute of Standards and Technology, Washington, DC, and Cambridge University Press, Cambridge, 2010. Available online in http://dlmf.nist.gov/10. [36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. [37] C. E. Pearson, Asymptotic behavior of solutions to the finite-difference wave equation, Math. Comp., 23 (1969), 711-715.  doi: 10.1090/S0025-5718-1969-0264862-4. [38] A. P. Prudnikov, A. Y. Brychkov and O. I. Marichev, Integrals and Series. Vol. 1. Elementary Functions Gordon and Breach Science Publishers, New York, 1986. [39] A. P. Prudnikov, A. Y. Brychkov and O. I. Marichev, Integrals and Series. Vol. 2. Special Functions Gordon and Breach Science Publishers, New York, 1990. [40] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces Princeton Univ. Press, 1971. [41] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680. [42] V. E. Tarasov and G. M. Zaslavsky, Fractional dynamics of coupled oscillators with long-range interaction, Chaos, 16 (2006), 023110, 13 pp. [43] V. E. Tarasov, Large lattice fractional Fokker-Planck equation, J. Stat. Mech., 2014 (2014). [44] V. E. Tarasov, Lattice fractional calculus, Appl. Math. Comp., 257 (2015), 12-33.  doi: 10.1016/j.amc.2014.11.033. [45] V. E. Tarasov, Fractional-order difference equations for physical lattices and some applications, J. Math. Phys. , 56 (2015), 103506, 19 pp. [46] V. E. Tarasov, Fractional Liouville equation on lattice phase-space, Phys. A, 421 (2015), 330-342.  doi: 10.1016/j.physa.2014.11.031. [47] E. S. Van Vleck and A. Zhang, Competing interactions and traveling wave solutions in lattice differential equations, Commun. Pure Appl. Anal., 15 (2016), 457-475.  doi: 10.3934/cpaa.2016.15.457. [48] J. B. Walsh, Estimating the time to the most recent common ancestor for the Y chromosome or mitochondrial DNA for a pair of individuals, Genetics, 158 (2001), 897-912. [49] G. A. Watson, A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. [50] K. Yosida, Functional Analysis reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. [51] L. Zhang and S. Guo, Existence and multiplicity of wave trains in 2D lattices, J. Differential Equations, 257 (2014), 759-783.  doi: 10.1016/j.jde.2014.04.016. [52] L. Zhou and W. Zhang, Admissibility and roughness of nonuniform exponential dichotomies for difference equations, J. Funct. Anal., 271 (2016), 1087-1129.  doi: 10.1016/j.jfa.2016.06.005.

show all references

##### References:
 [1] L. Abadías, M. de León-Contreras and J. L. Torrea, Non-local fractional derivatives. Discrete and continuous, J. Math. Anal. Appl., 449 (2017), 734-755.  doi: 10.1016/j.jmaa.2016.12.006. [2] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables National Bureau of Standards Applied Mathematics Series, no. 55, U. S. Government Printing Office, Washington, DC, 1964. [3] E. C. Aifantis, Continuum Nanomechanics for Nanocrystalline and Ultrafine Grain Materials Materials Science and Engineering, 63, 6th International Conference on Nanomaterials by Severe Plastic Deformation, 2014. doi: 10.1088/1757-899X/63/1/012129. [4] W. Arendt, C. Batty and S. Bu, Fourier multipliers for Hölder continuous functions and maximal regularity, Studia Math., 160 (2004), 23-51.  doi: 10.4064/sm160-1-2. [5] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems. Second edition. Monographs in Mathematics, 96. Birkhäuser/Springer Basel AG, Basel, 2011. xii+539 pp. [6] H. Bateman, Some simple differential difference equations and the related functions, Bull. Amer. Math. Soc., 49 (1943), 494-512.  doi: 10.1090/S0002-9904-1943-07927-X. [7] J. J. Betancor, A. J. Castro, J. C. Fariña and L. Rodríguez-Mesa, Discrete harmonic analysis associated with ultraspherical expansions, preprint, arXiv:1512.01379. [8] S. Bochner, Curvature and Betti numbers in Real and complex vector bundles, Univ. e Politec. di Torino. Rend. Sem. Mat., 15 (1955/56), 225-253. [9] J. Campbell, The SMM model as a boundary value problem using the discrete diffusion equation, Theor. Population Biol., 72 (2007), 539-546.  doi: 10.1016/j.tpb.2007.08.001. [10] S-N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478. [11] Ó. Ciaurri, T. A. Gillespie, L. Roncal, J. L. Torrea and J. L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math., 132 (2017), 109-131.  doi: 10.1007/s11854-017-0015-6. [12] Ó. Ciaurri, C. Lizama, L. Roncal and J. L. Varona, On a connection between the discrete fractional Laplacian and superdiffusion, Appl. Math. Letters, 49 (2015), 119-125.  doi: 10.1016/j.aml.2015.05.007. [13] Ó. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea and J. L. Varona, Fractional discrete Laplacian versus discretized fractional Laplacian, preprint, arXiv:1507.04986. [14] Ó. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea and J. L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, preprint, arXiv:1608.08913. [15] C. Corduneanu, Almost Periodic Functions Interscience Tracts in Pure and Applied Mathematics, New York-London-Sydney, 1968. [16] O. Defterli, M. D'Elia, Q. Du, M. Gunzburger, R. Lehoucq and M. M. Meerschaert, Fractional diffusion on bounded domains, Fract. Calc. Appl. Anal., 18 (2015), 342-360. [17] R. E. Edwards, Fourier Series: A Modern Introduction, Vol. 2, Second ed. , Springer-Verlag, 1982. [18] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194, Springer, New York, 2000. [19] G. Fath, Propagation failure of traveling waves in a discrete bistable medium, Phys. D, 116 (1998), 176-190.  doi: 10.1016/S0167-2789(97)00251-0. [20] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces North-Holland Mathematics Studies, 108. Notas de Matemática, 99. North-Holland Publishing Co. , Amsterdam, 1985. [21] A. Favini, R. Labbas, S. Maingot, H. Tanabe and A. Yagi, Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces, Discrete Contin. Dyn. Syst., 22 (2008), 973-987.  doi: 10.3934/dcds.2008.22.973. [22] W. Feller, An Introduction to Probability Theory and Its Applications. Vol. 1. Third edition. John Wiley & Sons, Inc. , New York-London-Sydney, 1968. [23] W. Feller, An Introduction to Probability Theory and Its Applications. Vol. 2. Second edition. John Wiley & Sons, Inc. , New York-London-Sydney, 1971. [24] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Second Edition Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer-Verlag, Berlin, 1983. [25] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Seventh Edition Elsevier Academic Press, New York, 2007. [26] F. A. Grünbaum, "The bispectral problem: An overview", in Special functions 2000: current perspective and future directions (Tempe, AZ), 129-140, NATO Sci. Ser. Ⅱ Math. Phys. Chem. 30, Kluwer Acad. Publ., Dordrecht, 2001. [27] F. A. Grünbaum and P. Iliev, Heat kernel expansions on the integers, Math. Phys. Anal. Geom., 5 (2002), 183-200.  doi: 10.1023/A:1016258207606. [28] J.-S. Guo and C.-C. Wu, The existence of traveling wave solutions for a bistable three-component lattice dynamical system, J. Differential Equations, 260 (2016), 1445-1455.  doi: 10.1016/j.jde.2015.09.036. [29] C. Hu and B. Li, Spatial dynamics for lattice differential equations with a shifting habitat, J. Differential Equations, 259 (2015), 1967-1989.  doi: 10.1016/j.jde.2015.03.025. [30] I. Lasiecka and M. Wilke, Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system, Discrete Contin. Dyn. Syst., 33 (2013), 5189-5202.  doi: 10.3934/dcds.2013.33.5189. [31] N. N. Lebedev, Special Functions and Its Applications Dover, New York, 1972. [32] J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dynam. Differential Equations, 11 (1999), 49-127.  doi: 10.1023/A:1021841618074. [33] J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, Dynamical Systems, Lecture Notes in Math., Springer, Berlin, 1822 (2003), 231-298. [34] C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators North-Holland Mathematics Studies, vol. 187, North-Holland Publishing Co. , Amsterdam, 2001. [35] F. W. J. Olver and L. C. Maximon, Bessel Functions, NIST handbook of mathematical functions (edited by F. W. F. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark), Chapter 10, National Institute of Standards and Technology, Washington, DC, and Cambridge University Press, Cambridge, 2010. Available online in http://dlmf.nist.gov/10. [36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. [37] C. E. Pearson, Asymptotic behavior of solutions to the finite-difference wave equation, Math. Comp., 23 (1969), 711-715.  doi: 10.1090/S0025-5718-1969-0264862-4. [38] A. P. Prudnikov, A. Y. Brychkov and O. I. Marichev, Integrals and Series. Vol. 1. Elementary Functions Gordon and Breach Science Publishers, New York, 1986. [39] A. P. Prudnikov, A. Y. Brychkov and O. I. Marichev, Integrals and Series. Vol. 2. Special Functions Gordon and Breach Science Publishers, New York, 1990. [40] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces Princeton Univ. Press, 1971. [41] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680. [42] V. E. Tarasov and G. M. Zaslavsky, Fractional dynamics of coupled oscillators with long-range interaction, Chaos, 16 (2006), 023110, 13 pp. [43] V. E. Tarasov, Large lattice fractional Fokker-Planck equation, J. Stat. Mech., 2014 (2014). [44] V. E. Tarasov, Lattice fractional calculus, Appl. Math. Comp., 257 (2015), 12-33.  doi: 10.1016/j.amc.2014.11.033. [45] V. E. Tarasov, Fractional-order difference equations for physical lattices and some applications, J. Math. Phys. , 56 (2015), 103506, 19 pp. [46] V. E. Tarasov, Fractional Liouville equation on lattice phase-space, Phys. A, 421 (2015), 330-342.  doi: 10.1016/j.physa.2014.11.031. [47] E. S. Van Vleck and A. Zhang, Competing interactions and traveling wave solutions in lattice differential equations, Commun. Pure Appl. Anal., 15 (2016), 457-475.  doi: 10.3934/cpaa.2016.15.457. [48] J. B. Walsh, Estimating the time to the most recent common ancestor for the Y chromosome or mitochondrial DNA for a pair of individuals, Genetics, 158 (2001), 897-912. [49] G. A. Watson, A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. [50] K. Yosida, Functional Analysis reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. [51] L. Zhang and S. Guo, Existence and multiplicity of wave trains in 2D lattices, J. Differential Equations, 257 (2014), 759-783.  doi: 10.1016/j.jde.2014.04.016. [52] L. Zhou and W. Zhang, Admissibility and roughness of nonuniform exponential dichotomies for difference equations, J. Funct. Anal., 271 (2016), 1087-1129.  doi: 10.1016/j.jfa.2016.06.005.
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