  We study the boundedness and convergence to equilibrium of weak solutions to
reaction-diffusion systems with nonlinear diffusion. The nonlinear diffusion is
of porous medium type and the nonlinear reaction terms are assumed to grow
polynomially and to dissipate (or conserve) the total mass. By utilising
duality estimates, the dissipation of the total mass and the smoothing effect
of the porous medium equation, we prove that if the exponents of the nonlinear
diffusion terms are high enough, then weak solutions are bounded, locally
H\"older continuous and their $L^{\infty}(\Omega)$-norm grows in time at most
polynomially.
  In order to show convergence to equilibrium, we consider a specific class of
nonlinear reaction-diffusion models, which describe a single reversible
reaction with arbitrarily many chemical substances. By exploiting a generalised
Logarithmic Sobolev Inequality, an indirect diffusion effect and the polynomial
in time growth of the $L^{\infty}(\Omega)$-norm, we show an entropy
entropy-production inequality which implies exponential convergence to
equilibrium in $L^p(\Omega)$-norm, for any $1\leq p < \infty$, with explicit
rates and constants.
