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-Hölder 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.