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 {\it 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. \'Ecole Norm. Sup. (4) 43 (2010), no. 1, 117--142] 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.