Young integrals and SPDEs

In this note, we study the non-linear evolution problem $dY_t = -A Y_t dt + B(Y_t) dX_t$, where $X$ is a $\gamma$-H\"older continuous function of the time parameter, with values in a distribution space, and $-A$ the generator of an analytical semigroup. Then, we will give some sharp conditions on $X$ in order to solve the above equation in a function space, first in the linear case (for any value of $\gamma$ in $(0,1)$), and then when $B$ satisfies some Lipschitz type conditions (for $\gamma>1/2$). The solution of the evolution problem will be understood in the mild sense, and the integrals involved in that definition will be of Young type.