Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian

We study the equations{document}align∂ₜ u(t, n) = L u(t, n) + f(u(t, n), n); ∂ₜ u(t, n) = iL u(t, n) + f(u(t, n), n)align{document}and{document}align∂ₜₜ u(t, n) =Lu(t, n) + f(u(t, n), n), align{document}where n∈ Z, $t∈ (0, ∞)$, and L is taken to be either the discrete Laplacian operator Δd f(n)=f(n+1)-2f(n)+f(n-1), or its fractional powers -(-Δd)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 Δd and -(-Δd)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 ZN are also accomplished.