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