Vol. 12, No. 7, 2019

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 6, 1871–2245
Issue 5, 1501–1870
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author index
To appear
Other MSP journals
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

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.

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