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

##### Abstract

We study the equations
$
\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)$
and
$
\partial_{tt} u(t,n) =Lu(t,n) + f(u(t,n),n)$, where $n\in \mathbb{Z}$, $t\in (0,\infty)$, and $L$ is taken to be either the discrete Laplacian operator $\Delta_{\operatorname{dis}} f(n)=f(n+1)-2f(n)+f(n-1)$, or its fractional powers $-(-\Delta_{\operatorname{dis}})^{\sigma}$, $0<\sigma<1$. We combine operator theory techniques with the properties of the Bessel functions to develop a theory of analytic semigroups and cosine operators generated by $\Delta_{\operatorname{dis}}$ and $-(-\Delta_{\operatorname{dis}})^{\sigma}$. 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 H\"older 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.