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

We study the equations\begin{document}$\begin{align}\partial_t u(t, n) = L u(t, n) + f(u(t, n), n); \partial_t u(t, n) = iL u(t, n) + f(u(t, n), n)\end{align}$\end{document}and\begin{document}$\begin{align}\partial_{tt} u(t, n) =Lu(t, n) + f(u(t, n), n), \end{align}$\end{document}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.