Reaction-Diffusion Equations with Transport Noise and Critical Superlinear Diffusion: Global Well-Posedness of Weakly Dissipative Systems

In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the d-dimensional torus. We show new global well-posedness results for a large class of scalar equations (e.g., the Allen-Cahn equation) and dissipative systems (e.g., equations in coagulation dynamics). Moreover, we prove global well-posedness for two weakly dissipative systems: Lotka-Volterra equations for d \in \{1, 2, 3, 4\} and the Brusselator for d \in \{1, 2, 3\}. Many of the results are also new without transport noise. The proofs are based on maximal regularity techniques, positivity results, and sharp blow-up criteria developed in our recent works, combined with energy estimates based on It\^ o's formula and stochastic Gronwall inequalities. Key novelties include the introduction of new L\zeta-coercivity/dissipativity conditions and the development of an Lp(Lq)framework for systems of reaction-diffusion equations, which are needed when treating dimensions d \in \{2, 3\} in the case of cubic or higher order nonlinearities.