Non-linear rough heat equations

This article is devoted to define and solve an evolution equation of the form dy _t ?=?Δy _t dt?+ dX _t (y _t ), where Δ stands for the Laplace operator on a space of the form ${L^p(mathbb R^n)}$ , and X is a finite dimensional noisy nonlinearity whose typical form is given by ${X_t(varphi)=sum_{i=1}^N , x^{i}_t f_i(varphi)}$ , where each x?=?(x ⁽¹⁾, … , x ^(N)) is a γ-H?lder function generating a rough path and each f _i is a smooth enough function defined on ${L^p(mathbb R^n)}$ . The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed.