Hölder?Lebesgue Regularity and Almost Periodicity for Semidiscrete Equations with a Fractional Laplacian

We study the equations ?tu(t, n) = Lu(t, n) + f(u(t, n), n); ?tu(t, n) = iLu(t, n) + f(u(t, n), n) and ?ttu(t, n) = Lu(t, n) + f(u(t, n), n), where n ? Z, t ? (0, ?), and L is taken to be either the discrete Laplacian operator ?df(n) = f(n+1)?2f(n)+f(n?1), or its fractional powers ?(??d)?, 0 < ? < 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 ?d and ?(??d)?. 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 Lapla-cians. The results obtained are applied to Nagumo and Fisher–KPP models with a discrete Laplacian. Further extensions to the multidimensional setting ZN are also accomplished.