  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.
