Vol. 12, No. 7, 2019

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Solutions of the 4-species quadratic reaction-diffusion system are bounded and $C^\infty$-smooth, in any space dimension

M. Cristina Caputo, Thierry Goudon and Alexis F. Vasseur

Vol. 12 (2019), No. 7, 1773–1804
Abstract

We establish the boundedness of solutions of reaction-diffusion systems with quadratic (in fact slightly superquadratic) reaction terms that satisfy a natural entropy dissipation property, in any space dimension N > 2. This bound implies the C-regularity of the solutions. This result extends the theory which was restricted to the two-dimensional case. The proof heavily uses De Giorgi’s iteration scheme, which allows us to obtain local estimates. The arguments rely on duality reasoning in order to obtain new estimates on the total mass of the system, both in the L(N+1)N norm and in a suitable weak norm. The latter uses Cα regularization properties for parabolic equations.

Keywords
reaction-diffusion systems, global regularity, blow-up methods
Mathematical Subject Classification 2010
Primary: 35K45, 35B65, 35K57
Milestones
Received: 28 November 2017
Revised: 22 August 2018
Accepted: 25 October 2018
Published: 22 July 2019
Authors
M. Cristina Caputo
Department of Mathematics
University of Texas at Austin
Austin, TX
United States
Thierry Goudon
Université Côte d’Azur, Inria, CNRS, LJAD
Nice
France
Alexis F. Vasseur
Department of Mathematics
University of Texas at Austin
Austin, TX
United States