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We study the boundedness and convergence to equilibrium of weak solutions toreaction-diffusion systems with nonlinear diffusion. The nonlinear diffusion isof porous medium type and the nonlinear reaction terms are assumed to growpolynomially and to dissipate (or conserve) the total mass. By utilisingduality estimates, the dissipation of the total mass and the smoothing effectof the porous medium equation, we prove that if the exponents of the nonlineardiffusion terms are high enough, then weak solutions are bounded, locallyH\"older continuous and their $L^{\infty}(\Omega)$-norm grows in time at mostpolynomially.  In order to show convergence to equilibrium, we consider a specific class ofnonlinear reaction-diffusion models, which describe a single reversiblereaction with arbitrarily many chemical substances. By exploiting a generalisedLogarithmic Sobolev Inequality, an indirect diffusion effect and the polynomialin time growth of the $L^{\infty}(\Omega)$-norm, we show an entropyentropy-production inequality which implies exponential convergence toequilibrium in $L^p(\Omega)$-norm, for any $1\leq p < \infty$, with explicitrates and constants.

Global regularity and convergence to equilibrium of reaction-diffusion systems with nonlinear diffusion