# 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 & Continuous Dynamical Systems, 2018, 38 (3) : 1365-1403. doi: 10.3934/dcds.2018056
##### References:

show all references

##### References:
 [1] Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 [2] Aijun Zhang. Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations. Conference Publications, 2013, 2013 (special) : 815-824. doi: 10.3934/proc.2013.2013.815 [3] Benjamin Contri. Fisher-KPP equations and applications to a model in medical sciences. Networks & Heterogeneous Media, 2018, 13 (1) : 119-153. doi: 10.3934/nhm.2018006 [4] François Hamel, James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik. A short proof of the logarithmic Bramson correction in Fisher-KPP equations. Networks & Heterogeneous Media, 2013, 8 (1) : 275-289. doi: 10.3934/nhm.2013.8.275 [5] Wenxian Shen, Zhongwei Shen. Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1193-1213. doi: 10.3934/cpaa.2016.15.1193 [6] Matt Holzer. A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2069-2084. doi: 10.3934/dcds.2016.36.2069 [7] Margarita Arias, Juan Campos, Cristina Marcelli. Fastness and continuous dependence in front propagation in Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 11-30. doi: 10.3934/dcdsb.2009.11.11 [8] Jean-Michel Roquejoffre, Luca Rossi, Violaine Roussier-Michon. Sharp large time behaviour in $N$-dimensional Fisher-KPP equations. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7265-7290. doi: 10.3934/dcds.2019303 [9] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5023-5045. doi: 10.3934/dcdsb.2020323 [10] Denis Pennequin. Existence of almost periodic solutions of discrete time equations. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 51-60. doi: 10.3934/dcds.2001.7.51 [11] Luca Lorenzi. Optimal Hölder regularity for nonautonomous Kolmogorov equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 169-191. doi: 10.3934/dcdss.2011.4.169 [12] Denghui Wu, Zhen-Hui Bu. Multidimensional stability of pyramidal traveling fronts in degenerate Fisher-KPP monostable and combustion equations. Electronic Research Archive, 2021, 29 (6) : 3721-3740. doi: 10.3934/era.2021058 [13] Grégory Faye, Thomas Giletti, Matt Holzer. Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021146 [14] Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157 [15] Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121 [16] Matthieu Alfaro, Arnaud Ducrot. Sharp interface limit of the Fisher-KPP equation. Communications on Pure & Applied Analysis, 2012, 11 (1) : 1-18. doi: 10.3934/cpaa.2012.11.1 [17] Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703 [18] Fang Li, Xing Liang, Wenxian Shen. Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317 [19] Angelo Favini, Rabah Labbas, Stéphane Maingot, Hiroki Tanabe, Atsushi Yagi. Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 973-987. doi: 10.3934/dcds.2008.22.973 [20] Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1857-1882. doi: 10.3934/dcds.2013.33.1857

2020 Impact Factor: 1.392