An integrable bound for semilinear rough partial differential equations with unbounded diffusion coefficients

This work develops moment bounds for the controlled rough path norm of the solution of semilinear rough partial differential equations.~The novel aspects are two-fold: first we consider rough paths of low time regularity $\gamma \in (1\mathrm{/}4,1\mathrm{/}2)$ and second treat unbounded diffusion coefficients. To this aim we introduce a suitable notion of a controlled rough path according to a monotone scale of Banach spaces and innovative control functions.