article published yes Antonio Agresti author 673cd0cc-9b9a-11eb-b144-88f30e1fbb720000-0002-9573-2962 Mark Veraar author JuFi department Bridging Scales in Random Materials project In this paper we consider a class of stochastic reaction-diffusion equations. We provide local well-posedness, regularity, blow-up criteria and positivity of solutions. The key novelties of this work are related to the use transport noise, critical spaces and the proof of higher order regularity of solutions – even in case of non-smooth initial data. Crucial tools are Lp(Lp)-theory, maximal regularity estimates and sharp blow-up criteria. We view the results of this paper as a general toolbox for establishing global well-posedness for a large class of reaction-diffusion systems of practical interest, of which many are completely open. In our follow-up work [8], the results of this paper are applied in the specific cases of the Lotka-Volterra equations and the Brusselator model. https://research-explorer.ista.ac.at/download/13135/14895/2023_JourDifferentialEquations_Agresti.pdf application/pdfno Elsevier2023 eng 0022-0396 1090-2732 00101901870000110.1016/j.jde.2023.05.038 3689247-300 Agresti A, Veraar M. Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity. <i>Journal of Differential Equations</i>. 2023;368(9):247-300. doi:<a href="https://doi.org/10.1016/j.jde.2023.05.038">10.1016/j.jde.2023.05.038</a> Agresti, Antonio, and Mark Veraar. “Reaction-Diffusion Equations with Transport Noise and Critical Superlinear Diffusion: Local Well-Posedness and Positivity.” <i>Journal of Differential Equations</i>. Elsevier, 2023. <a href="https://doi.org/10.1016/j.jde.2023.05.038">https://doi.org/10.1016/j.jde.2023.05.038</a>. Agresti, Antonio, and Mark Veraar. “Reaction-Diffusion Equations with Transport Noise and Critical Superlinear Diffusion: Local Well-Posedness and Positivity.” <i>Journal of Differential Equations</i>, vol. 368, no. 9, Elsevier, 2023, pp. 247–300, doi:<a href="https://doi.org/10.1016/j.jde.2023.05.038">10.1016/j.jde.2023.05.038</a>. Agresti A, Veraar M. 2023. Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity. Journal of Differential Equations. 368(9), 247–300. A. Agresti, M. Veraar, Journal of Differential Equations 368 (2023) 247–300. Agresti, A., &#38; Veraar, M. (2023). Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity. <i>Journal of Differential Equations</i>. Elsevier. <a href="https://doi.org/10.1016/j.jde.2023.05.038">https://doi.org/10.1016/j.jde.2023.05.038</a> A. Agresti and M. Veraar, “Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity,” <i>Journal of Differential Equations</i>, vol. 368, no. 9. Elsevier, pp. 247–300, 2023. 131352023-06-18T22:00:45Z2025-04-14T07:53:59Z