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

2017/11/08 by Klemens Fellner, Fellner, Klemens, Evangelos Latos +3
Mathematics · Physics and Astronomy · Engineering · #Mathematical Biology Tumor Growth #Advanced Thermodynamics and Statistical Mechanics #Phase Equilibria and Thermodynamics

paper · pdf · doi:10.48550/arxiv.1711.02897

Abstract

We study the boundedness and convergence to equilibrium of weak solutions to\nreaction-diffusion systems with nonlinear diffusion. The nonlinear diffusion is\nof porous medium type and the nonlinear reaction terms are assumed to grow\npolynomially and to dissipate (or conserve) the total mass. By utilising\nduality estimates, the dissipation of the total mass and the smoothing effect\nof the porous medium equation, we prove that if the exponents of the nonlinear\ndiffusion terms are high enough, then weak solutions are bounded, locally\nH "older continuous and their L^\∞(\Ω)-norm grows in time at most\npolynomially.\n In order to show convergence to equilibrium, we consider a specific class of\nnonlinear reaction-diffusion models, which describe a single reversible\nreaction with arbitrarily many chemical substances. By exploiting a generalised\nLogarithmic Sobolev Inequality, an indirect diffusion effect and the polynomial\nin time growth of the L^\∞(\Ω)-norm, we show an entropy\nentropy-production inequality which implies exponential convergence to\nequilibrium in L^p(\Ω)-norm, for any 1\≤ p < \∞, with explicit\nrates and constants.\n

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