Indirect diffusion effect in degenerate reaction-diffusion systems

Abstract

In this work we study global well-posedness and large time behaviour for a typical reaction-diffusion system, whose non-linearities arise from chemical reactions, and which include degenerate diffusion. We show that there is an indirect diffusion effect, i.e. an effective diffusion for the non-diffusive species which is incurred by a combination of diffusion from diffusive species and reversible reactions between the species. Deriving new estimates for such degenerate reaction-diffusion system, we show, by applying the entropy method, that the solution converges exponentially to equilibrium, and provide explicit convergence rates and associated constants. Contents 1. Introduction 1 1.1. Reaction-diffusion equations 1 1.2. The setting of the problem 2 1.3. The current state of the art 4 1.4. Main results and key ideas 5 2. Indirect diffusion effect and the entropy method 9 2.1. The entropic inequalities 9 2.2. Conditional convergence to equilibrium 18 3. Global existence of solutions 20 3.1. Auxiliary results 21 3.2. Proof of Proposition 2 24 3.3. Proof of Proposition 1 37 4. The interaction between the entropy and the norm bounds: Proof of the main theorems 38 5. Final Remarks 43 References 43