Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion

TL;DR

This paper proves the global existence and exponential convergence to equilibrium of renormalised solutions for reaction-diffusion systems with non-linear diffusion, using entropy methods and weighted truncation functions.

Contribution

It introduces a new approach to establish global solutions for degenerate non-linear diffusion systems and analyzes their large-time behaviour in complex balanced reaction networks.

Findings

Global renormalised solutions exist for the systems studied.

Solutions converge exponentially to equilibrium when no boundary equilibria are present.

Convergence results extend to certain non-linear diffusions where existence was previously unknown.

Abstract

The global existence of renormalised solutions and convergence to equilibrium for reaction-diffusion systems with non-linear diffusion are investigated. The system is assumed to have quasi-positive non-linearities and to satisfy an entropy inequality. The difficulties in establishing global renormalised solutions caused by possibly degenerate diffusion are overcome by introducing a new class of weighted truncation functions. By means of the obtained global renormalised solutions, we study the large-time behaviour of complex balanced systems arising from chemical reaction network theory with non-linear diffusion. When the reaction network does not admit boundary equilibria, the complex balanced equilibrium is shown, by using the entropy method, to exponentially attract all renormalised solutions in the same compatibility class. This convergence extends even to a range of non-linear diffusion, where global existence is an open problem, yet we are able to show that solutions to approximate systems converge exponentially to equilibrium uniformly in the regularisation parameter.