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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 |
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Vol. 12 (2019), No. 7, 1773–1804
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Abstract |
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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 . This bound implies the -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 norm and in a suitable weak norm. The latter uses regularization properties for parabolic equations. |
Keywords
reaction-diffusion systems, global regularity, blow-up
methods
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Mathematical Subject Classification 2010
Primary: 35K45, 35B65, 35K57
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Milestones
Received: 28 November 2017
Revised: 22 August 2018
Accepted: 25 October 2018
Published: 22 July 2019
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Authors |
| M. Cristina Caputo | |
| Department of Mathematics University of Texas at Austin Austin, TX United States |
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| Thierry Goudon | |
| Université Côte d’Azur, Inria, CNRS,
LJAD Nice France |
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| Alexis F. Vasseur | |
| Department of Mathematics University of Texas at Austin Austin, TX United States |
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