Time-step heat problem on the mesh: asymptotic behavior and decay rates

References

[1] A. Abadías and E. Alvarez, Asymptotic behavior for the discrete in time heat equation, Mathematics 10 (2022), no. 17, Paper No. 3128. 10.3390/math10173128Search in Google Scholar

[2] L. Abadias and M. De León-Contreras, Discrete Hölder spaces and their characterization via semigroups associated with the discrete Laplacian and kernel estimates, J. Evol. Equ. 22 (2022), no. 4, Paper No. 91. 10.1007/s00028-022-00851-1Search in Google Scholar

[3] L. Abadias, M. De León-Contreras and J. L. Torrea, Non-local fractional derivatives. Discrete and continuous, J. Math. Anal. Appl. 449 (2017), no. 1, 734–755. 10.1016/j.jmaa.2016.12.006Search in Google Scholar

[4] L. Abadías, J. González Camus, P. J. Miana and J. C. Pozo, Large time behaviour for the heat equation on ℤ ; moments and decay rates, J. Math. Anal. Appl. 500 (2021), no. 2, Article ID 125137. 10.1016/j.jmaa.2021.125137Search in Google Scholar

[5] R. P. Agarwal, C. Cuevas and C. Lizama, Regularity of Difference Equations on Banach Spaces, Springer, Cham, 2014. 10.1007/978-3-319-06447-5Search in Google Scholar

[6] D. R. Anderson, R. I. Avery and J. M. Davis, Existence and uniqueness of solutions to discrete diffusion equations, Comput. Math. Appl. 45 (2003), 1075–1085. 10.1016/S0898-1221(03)00087-7Search in Google Scholar

[7] H. Bateman, Some simple differential difference equations and the related functions, Bull. Amer. Math. Soc. 49 (1943), 494–512. 10.1090/S0002-9904-1943-07927-XSearch in Google Scholar

[8] A. Benchellal, T. Poinot and J.-C. Trigeassou, Approximation and identification of diffusive interfaces by fractional models, Signal Process. 86 (2006), no. 10, 2712–2727. 10.1016/j.sigpro.2006.02.025Search in Google Scholar

[9] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. Search in Google Scholar

[10] S. Blunck, Analyticity and discrete maximal regularity on L p -spaces, J. Funct. Anal. 183 (2001), no. 1, 211–230. 10.1006/jfan.2001.3740Search in Google Scholar

[11] S. Blunck, Maximal regularity of discrete and continuous time evolution equations, Studia Math. 146 (2001), no. 2, 157–176. 10.4064/sm146-2-3Search in Google Scholar

[12] O. 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. 10.1007/s11854-017-0015-6Search in Google Scholar

[13] O. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea and J. L. Varona, Nonlocal discrete diffusion equations and the fractional discreteLaplacian, regularity and applications, Adv. Math. 330 (2018), 688–738. 10.1016/j.aim.2018.03.023Search in Google Scholar

[14] E. B. Davies, Gaussian upper bounds for the heat kernels of some second-order operators on Riemannian manifolds, J. Funct. Anal. 80 (1988), no. 1, 16–32. 10.1016/0022-1236(88)90062-6Search in Google Scholar

[15] E. B. Davies, L p spectral theory of higher-order elliptic differential operators, Bull. Lond. Math. Soc. 29 (1997), no. 5, 513–546. 10.1112/S002460939700324XSearch in Google Scholar

[16] M. Del Pino and J. Dolbeault, Asymptotic behavior of nonlinear diffusions, Math. Res. Lett. 10 (2003), no. 4, 551–557. 10.4310/MRL.2003.v10.n4.a13Search in Google Scholar

[17] J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 6, 693–698. Search in Google Scholar

[18] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math. 194, Springer, New York, 1980. Search in Google Scholar

[19] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. Vol. I, McGraw-Hill, New York, 1953. Search in Google Scholar

[20] M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in 𝐑 N , J. Funct. Anal. 100 (1991), no. 1, 119–161. 10.1016/0022-1236(91)90105-ESearch in Google Scholar

[21] E. Estrada, E. Hameed, N. Hatano and M. Langer, Path Laplacian operators and superdiffusive processes on graphs. I. One-dimensional case, Linear Algebra Appl. 523 (2017), 307–334. 10.1016/j.laa.2017.02.027Search in Google Scholar

[22] S. N. Ethier and T. G. Kurtz, Fleming–Viot processes in population genetics, SIAM J. Control Optim. 31 (1993), no. 2, 345–386. 10.1137/0331019Search in Google Scholar

[23] W. Feller, An Introduction to Probability Theory and its Applications. Vol. II, 2nd ed., John Wiley & Sons, New York, 1971. Search in Google Scholar

[24] G. B. Folland, Introduction to Partial Differential Equations, 2nd ed., Princeton University, Princeton, 1995. Search in Google Scholar

[25] J. B. J. Fourier, Théorie analytique de la chaleur, Cambridge Libr. Coll., Cambridge University, Cambridge, 2009. 10.1017/CBO9780511693229Search in Google Scholar

[26] A. Gmira and L. Véron, Asymptotic behaviour of the solution of a semilinear parabolic equation, Monatsh. Math. 94 (1982), no. 4, 299–311. 10.1007/BF01667384Search in Google Scholar

[27] A. Gmira and L. Véron, Large time behaviour of the solutions of a semilinear parabolic equation in 𝐑 N , J. Differential Equations 53 (1984), no. 2, 258–276. 10.1016/0022-0396(84)90042-1Search in Google Scholar

[28] J. González-Camus, V. Keyantuo, C. Lizama and M. Warma, Fundamental solutions for discrete dynamical systems involving the fractional Laplacian, Math. Methods Appl. Sci. 42 (2019), no. 14, 4688–4711. 10.1002/mma.5685Search in Google Scholar

[29] J. González-Camus, C. Lizama and P. J. Miana, Fundamental solutions for semidiscrete evolution equations via Banach algebras, Adv. Difference Equ. 2021 (2021), Paper No. 35. 10.1186/s13662-020-03206-7Search in Google Scholar PubMed PubMed Central

[30] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Search in Google Scholar

[31] Y. Jiang, X. Wang and Y. Wang, On a stochastic heat equation with first order fractional noises and applications to finance, J. Math. Anal. Appl. 396 (2012), no. 2, 656–669. 10.1016/j.jmaa.2012.07.003Search in Google Scholar

[32] T. Kemmochi, Discrete maximal regularity for abstract Cauchy problems, Studia Math. 234 (2016), no. 3, 241–263. 10.4064/sm8495-7-2016Search in Google Scholar

[33] V. Keyantuo, C. Lizama and M. Warma, Lattice dynamical systems associated with a fractional Laplacian, Numer. Funct. Anal. Optim. 40 (2019), no. 11, 1315–1343. 10.1080/01630563.2019.1602542Search in Google Scholar

[34] S. Kusuoka and D. Stroock, Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator, Ann. of Math. (2) 127 (1988), no. 1, 165–189. 10.2307/1971418Search in Google Scholar

[35] N. N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972. Search in Google Scholar

[36] P. Li, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. of Math. (2) 124 (1986), no. 1, 1–21. 10.2307/1971385Search in Google Scholar

[37] C. Lizama and M. Murillo-Arcila, On a connection between the N-dimensional fractional Laplacian and 1-D operators on lattices, J. Math. Anal. Appl. 511 (2022), no. 1, Paper No. 126051. 10.1016/j.jmaa.2022.126051Search in Google Scholar

[38] C. Lizama and L. Roncal, Hölder–Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian, Discrete Contin. Dyn. Syst. 38 (2018), no. 3, 1365–1403. 10.3934/dcds.2018056Search in Google Scholar

[39] S. Mustapha, Gaussian estimates for heat kernels on Lie groups, Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 1, 45–64. 10.1017/S0305004199003783Search in Google Scholar

[40] J. R. Norris, Long-time behaviour of heat flow: global estimates and exact asymptotics, Arch. Ration. Mech. Anal. 140 (1997), no. 2, 161–195. 10.1007/s002050050063Search in Google Scholar

[41] F. Oberhettinger and L. Badii, Tables of Laplace Transforms, Springer, New York, 1973. 10.1007/978-3-642-65645-3Search in Google Scholar

[42] A. Slavík, Discrete Bessel functions and partial difference equations, J. Difference Equ. Appl. 24 (2018), no. 3, 425–437. 10.1080/10236198.2017.1416107Search in Google Scholar

[43] A. Slavík, Asymptotic behavior of solutions to the semidiscrete diffusion equation, Appl. Math. Lett. 106 (2020), Article ID 106392. 10.1016/j.aml.2020.106392Search in Google Scholar

[44] A. Slavík, Spatial maxima, unimodality, and asymptotic behaviour of solutions to discrete diffusion-type equations, J. Difference Equ. Appl. 28 (2022), no. 1, 126–140. 10.1080/10236198.2022.2027395Search in Google Scholar

[45] A. Slavík and P. Stehlík, Dynamic diffusion-type equations on discrete-space domains, J. Math. Anal. Appl. 427 (2015), no. 1, 525–545. 10.1016/j.jmaa.2015.02.056Search in Google Scholar

[46] P. Stehlík and J. Volek, Nonuniqueness of implicit lattice Nagumo equation, Appl. Math. 64 (2019), no. 2, 169–194. 10.21136/AM.2019.0270-18Search in Google Scholar

[47] F. G. Tricomi and A. Erdélyi, The asymptotic expansion of a ratio of gamma functions, Pacific J. Math. 1 (1951), 133–142. 10.2140/pjm.1951.1.133Search in Google Scholar

[48] J. L. Vázquez, Asymptotic behaviour methods for the heat equation. Convergence to the Gaussian, preprint (2017), https://arxiv.org/abs/1706.10034. Search in Google Scholar

[49] E. Zuazua, Large time asymptotics for heat and dissipative wave equations, preprint (2003), https://www.researchgate.net/publication/228560600. Search in Google Scholar

[50] A. Zygmund, Trigonometric Series.Vols. I, II, 2nd ed., Cambridge University, New York, 1959. Search in Google Scholar