The Entropy Dissipation Method for Spatially Inhomogeneous Reaction–diffusion Type Systems

We study the large–time asymptotics of reaction–diffusion type systems, which feature a monotone decaying entropy (Lyapunov, free energy) functional. We consider both bounded domains and confining potentials on the whole space for arbitrary space dimensions. Our aim is to derive quantitative expressions for (or estimates of) the rates of convergence towards an (entropy minimising) equilibrium state in terms of the constants of diffusion and reaction and with respect to conserved quantities. Our method, the so called entropy approach, seeks to quantify convergence to equilibrium by using functional inequalities, which relate quantitatively the entropy and its dissipation in time. The entropy approach is well suited to nonlinear problems and known to be quite robust with respect to model variations. It has already been widely applied to scalar diffusion–convection equations and the main goal of this paper is to study its generalisation to systems of partial differential equations, which contain diffusion and reaction terms and admit fewer conservation laws than the size of the system. In particular, we successfully apply the entropy approach to general linear systems and to a nonlinear example of reaction–diffusion–convection system arising in solid state physics as a paradigm for general nonlinear systems.