  Combining fractional calculus and the Rough Path Theory we study the
existence and uniqueness of mild solutions to evolutions equations driven by a
H\"older continuous function with H\"older exponent in $(1/3,1/2)$. Our
stochastic integral is in some sense a generalization of the well-known Young
integral and can be defined independently of the initial condition. Similar to
the Rough Path Theory we establish a second variable which is given, roughly
speaking, by a tensor product. It is then necessary to formulate a second
equation for this new variable, and we do in a mild sense. The crucial point in
order to get this new equation is to construct a tensor depending on the noise
path but also on the semigroup. We then prove the existence of a unique
H\"older continuous solution of the system of equations, consisting of the path
and the area components, if the nonlinear term and the initial condition are
sufficiently smooth. Once the abstract theory is developed, we can present a
pathwise nonlinear SPDE driven by a fractional Brownian motion with Hurst
parameter in $(1/3,1/2]$.
