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

Abstract: 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.