Global solutions for semilinear rough partial differential equations

The paper proves global-in-time well-posedness for semilinear parabolic RPDEs dy_t = [A y_t + F(y_t)] dt + G(y_t) dX_t on a scale of interpolation spaces, under (F) linear growth/Lipschitz and (G) bounded C^3 with a bounded derivative of DG(·)G(·), by combining local existence from prior work with new linear a-priori bounds that avoid quadratic growth and a concatenation argument (Theorem 3.9, supported by Lemmas 3.5–3.6 and Corollary 3.7) . The candidate solution follows the same controlled rough path framework, constructs the rough convolution with the analytic semigroup, proves local well-posedness via a contraction on the graph-of-G, and extends globally using the same type of a-priori estimates; differences are methodological (the paper cites a prior local theory, the model rederives a contraction). No substantive contradictions were found.