A one-dimensional non-local singular SPDE

We examine in this article the one-dimensional, non-local, singular SPDE begin{equation*} partial_t u ;=; -, (-Delta)^{1/2} u ,-, sinh(gamma u) ,+, xi;, end{equation*} where $gammain mathbb{R}$, $(-Delta)^{1/2}$ is the fractional Laplacian of order $1/2$, $xi$ the space-time white noise in $mathbb{R} times mathbb{T}$, and $mathbb{T}$ the one-dimensional torus. We show that for $0<gamma^2<pi/7$ the Da Prato--Debussche method applies. One of the main difficulties lies in the derivation of a Schauder estimate for the semi-group associated to the fractional Laplacian due to the lack of smoothness resulting from the long range interaction.