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We investigate existence, uniqueness and regularity for local solutions ofrough parabolic equations with subcritical noise of the form $du_t- L_tu_tdt=N(u_t)dt + \sum_{i = 1}^dF_i(u_t)d\mathbf X^i_t$ where $(L_t)_{t\in[0,T]}$ is atime-dependent family of unbounded operators acting on some scale of Banachspaces, while $\mathbf X\equiv(X,\mathbb X)$ is a two-step (non-necessarilygeometric) rough path of H\"older regularity $\gamma >1/3.$ Besides dealingwith non-autonomous evolution equations, our results also allow for unboundedoperations in the noise term (up to some critical loss of regularity dependingon that of the rough path $\mathbf X$). As a technical tool, we introduce aversion of the multiplicative sewing lemma, which allows to construct theso-called product integrals in infinite dimensions. We later use it toconstruct a semigroup analogue for the non-autonomous linear PDEs as well asshow how to deduce the semigroup version of the usual sewing lemma from it.

Non-autonomous rough semilinear PDEs and the multiplicative Sewing Lemma