Global classical solutions to reaction-diffusion systems in one and two dimensions

The global existence of classical solutions to reaction-diffusion systems in dimensions one and two is proved. The considered systems are assumed to satisfy an entropy inequality and have nonlinearities with at most cubic growth in 1D or at most quadratic growth in 2D. This global existence was already proved in [T. Goudon and A. Vasseur, Ann. Sci. École Norm. Sup., 43(1):117–142, 2010] by a De Giorgi method. In this paper, we give a simplified proof by using a modified Gagliardo–Nirenberg inequality and the regularity of the heat operator. Moreover, the classical solution is proved to have $L^{\infty}$-norm growing at most polynomially in time. As an application, solutions to chemical reaction-diffusion systems satisfying the so-called complex balance condition are proved to converge exponentially to equilibrium in $L^{\infty}$-norm.

Keywords

reaction-diffusion systems, global classical solutions, entropy estimates, chemical reaction networks

2010 Mathematics Subject Classification

35B40, 35K57, 35Q92, 80A30, 80A32

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The author would like to thank Prof. Laurent Desvillettes and Prof. Klemens Fellner for fruitful discussion, which leads to this work. This work is partially supported by International Training Program IGDK 1754 and NAWI Graz.

Received 24 August 2017

Accepted 22 November 2017

Published 14 May 2018