Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity

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...

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Bibliographic Details
Published in:Journal of Differential Equations 2023-09, Vol.368, p.247-300
Main Authors: Agresti, Antonio, Veraar, Mark
Format: Article
Language:English
Subjects:
Critical spaces
Local and global well-posedness
Positivity
Reaction-diffusion equations
Stochastic partial differential equations
Transport noise
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Summary: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(Lq)-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.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2023.05.038