Subordination principle, Wright functions and large-time behavior for the discrete in time fractional diffusion equation. (English) Zbl 1478.35029

Summary: The main goal in this paper is to study asymptotic behavior in \(L^p( \mathbb{R}^N)\) for the solutions of the fractional version of the discrete in time \(N\)-dimensional diffusion equation, which involves the Caputo fractional \(h\)-difference operator. The techniques to prove the results are based in new subordination formulas involving the discrete in time Gaussian kernel, and which are defined via an analogue in discrete time setting of the scaled Wright functions. Moreover, we get an equivalent representation of that subordination formula by Fox H-functions.


35B40 Asymptotic behavior of solutions to PDEs
35R11 Fractional partial differential equations
35K15 Initial value problems for second-order parabolic equations
39A12 Discrete version of topics in analysis
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[1] Abadias, L.; Alvarez, E., Uniform stability for fractional Cauchy problems and applications, Topol. Methods Nonlinear Anal., 52, 2, 707-728 (2018) · Zbl 1414.34003
[2] Abadias, L.; Alvarez, E., Asymptotic behaviour for the discrete in time heat equation, Manuscript available at
[3] Abadias, L.; De León, M.; Torrea, J. L., Non-local fractional derivatives. Discrete and continuous, J. Math. Anal. Appl., 449, 1, 734-755 (2017) · Zbl 1355.26004
[4] Abadias, L.; Lizama, C., Almost automorphic mild solutions to fractional partial difference-differential equations, Appl. Anal., 95, 6, 1347-1369 (2016) · Zbl 1381.35205
[5] Abadias, L.; Miana, P., A subordination principle on Wright functions and regularized resolvent families, J. Funct. Spaces, 2015, Article 158145 pp. (2015) · Zbl 1354.47028
[6] Aigner, M., Diskrete Mathematik (2006), Friedr. Vieweg & Sohn · Zbl 1109.05001
[7] Alvarez, E.; Diaz, S.; Lizama, C., C-semigroups, subordination principle and the Lévy α-stable distribution on discrete time, Commun. Contemp. Math. (2020)
[8] Bazhlekova, E. G., Fractional evolution equations in Banach spaces (2001), University Press Facilities, Eindhoven University of Technology, Ph.D. thesis
[9] Ciaurri, O.; Gillespie, T. A.; Roncal, L.; Torrea, J. L.; Varona, J. L., Harmonic analysis associated with a discrete Laplacian, J. Anal. Math., 132, 109-131 (2017) · Zbl 1476.39003
[10] Ciaurri, O.; Roncal, L.; Stinga, P. R.; Torrea, J. L.; Varona, J. L., Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math., 330, 688-738 (2018) · Zbl 1391.35388
[11] Davies, E. B., Gaussian upper bounds for the heat kernels of some second-order operators on Riemannian manifolds, J. Funct. Anal., 80, 1, 16-32 (1988) · Zbl 0759.58045
[12] Davies, E. B., \( L^p\) spectral theory of higher-order elliptic differential operators, Bull. Lond. Math. Soc., 29, 5, 513-546 (1997) · Zbl 0955.35019
[13] Del Pino, M.; Dolbeault, J., Asymptotic behavior of nonlinear diffusions, Math. Res. Lett., 10, 4, 551-557 (2003) · Zbl 1045.35009
[14] Duoandikoetxea, J.; Zuazua, J., Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. I Math., 315, 6, 693-698 (1992) · Zbl 0755.45019
[15] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G.; Bateman, H., Higher Transcenden-tal Functions, vol. III (1953), McGraw-Hill: McGraw-Hill New York
[16] Erdélyi, A.; Tricomi, F. G., The aymptotic expansion of a ratio of Gamma functions, Pac. J. Math., 1, 133-142 (1951) · Zbl 0043.29103
[17] Escobedo, M.; Zuazua, E., Large time behavior for convection-diffusion equations in \(\mathbb{R}^N\), J. Funct. Anal., 100, 1, 119-161 (1991) · Zbl 0762.35011
[18] Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics, vol. 19 (2014), AMS Publications: AMS Publications Providence, Rhode Island
[19] Fourier, J., Théorie Analytique de la Chaleur, Cambridge Library Collection (2009), Cambridge University Press: Cambridge University Press Cambridge, Reprint of the 1822 original · JFM 15.0954.01
[20] Gmira, A.; Veron, L., Asymptotic behaviour of the solution of a semilinear parabolic equation, Monatshefte Math., 94, 299-311 (1982) · Zbl 0502.35014
[21] Gmira, A.; Veron, L., Large time behaviour of the solutions of a semilinear parabolic equation in \(\mathbb{R}^N\), J. Funct. Anal., 53, 258-276 (1984) · Zbl 0529.35041
[22] Goodrich, C.; Lizama, C., A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity, Isr. J. Math., 236, 533-589 (2020) · Zbl 07202576
[23] Goodrich, C.; Peterson, A. C., Discrete Fractional Calculus (2015), Springer International Publishing · Zbl 1350.39001
[24] Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products (2000), Academic Press, Inc.: Academic Press, Inc. San Diego, CA · Zbl 0981.65001
[25] Grigor’yan, A., Estimates of heat kernels on Riemannian manifolds (1999), manuscript available at · Zbl 0985.58007
[26] Kemppainen, J.; Siljander, J.; Zacher, R., Representation of solutions and large-time behavior for fully nonlocal diffusion equations, J. Differ. Equ., 263, 1, 149-201 (2017) · Zbl 1366.35218
[27] Kilbas, A. A.; Saigo, M., H-Transforms, Theory and Applications, Analytical Methods and Special Functions, vol. 9 (2004) · Zbl 1056.44001
[28] Kusuoka, S.; Stroock, D., Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator, Ann. Math., 127, 1, 165-189 (1988) · Zbl 0699.35025
[29] Li, P., Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. Math., 124, 1, 1-21 (1986) · Zbl 0613.58032
[30] Lizama, C., lp-Maximal regularity for fractional difference equations on UMD spaces, Math. Nachr., 288, 17/18, 2079-2092 (2015) · Zbl 1335.39003
[31] Lizama, C., The Poisson distribution, abstract fractional difference equations and stability, Proc. Am. Math. Soc., 145, 9, 3809-3827 (2017) · Zbl 1368.39001
[32] Lizama, C.; Roncal, L., Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian, Discrete Contin. Dyn. Syst., 38, 3, 1365-1403 (2018) · Zbl 1397.34034
[33] Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (2010), Imperial College Press: Imperial College Press London, UK · Zbl 1210.26004
[34] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley & Sons: John Wiley & Sons New York, NY, USA · Zbl 0789.26002
[35] Mozyrska, D.; Wyrwas, M., The Z-transform method and delta type fractional difference operators, Discrete Dyn. Nat. Soc., 2015, Article 852734 pp. (2015) · Zbl 1418.44003
[36] Mustapha, S., Gaussian estimates for heat kernels on Lie groups, Math. Proc. Camb. Philos. Soc., 128, 1, 45-64 (2000) · Zbl 0947.22007
[37] Norris, J. R., Long-time behaviour of heat flow: global estimates and exact asymptotics, Arch. Ration. Mech. Anal., 140, 2, 161-195 (1997) · Zbl 0899.35015
[38] Ponce, R., Time discretization of fractional subdiffusion equations via fractional resolvent operators, Comput. Math. Appl., 80, 4, 69-92 (2020) · Zbl 1446.65075
[39] Zygmund, A., Trigonometric Series, Vols. I, II (1959), Cambridge University Press: Cambridge University Press New York
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