Global Renormalised Solutions and Equilibration of Reaction–Diffusion Systems with Nonlinear Diffusion

Abstract

The global existence of renormalised solutions and convergence to equilibrium for reaction–diffusion systems with nonlinear diffusion are investigated. The system is assumed to have quasi-positive nonlinearities 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 nonlinear diffusion. When the reaction network does not admit boundary equilibria, the complex balanced equilibrium is shown, by using the entropy method, to exponentially attract renormalised solutions in the same compatibility class. This convergence extends even to a range of nonlinear 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.