Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction–diffusion systems

  • Klemens FellnerEmail author
  • Bao Quoc Tang
Open Access


The convergence to equilibrium for renormalised solutions to nonlinear reaction–diffusion systems is studied. The considered reaction–diffusion systems arise from chemical reaction networks with mass action kinetics and satisfy the complex balanced condition. By applying the so-called entropy method, we show that if the system does not have boundary equilibria, i.e. equilibrium states lying on the boundary of \({\mathbb {R}}_+^N\), then any renormalised solution converges exponentially to the complex balanced equilibrium with a rate, which can be computed explicitly up to a finite-dimensional inequality. This inequality is proven via a contradiction argument and thus not explicitly. An explicit method of proof, however, is provided for a specific application modelling a reversible enzyme reaction by exploiting the specific structure of the conservation laws. Our approach is also useful to study the trend to equilibrium for systems possessing boundary equilibria. More precisely, to show the convergence to equilibrium for systems with boundary equilibria, we establish a sufficient condition in terms of a modified finite-dimensional inequality along trajectories of the system. By assuming this condition, which roughly means that the system produces too much entropy to stay close to a boundary equilibrium for infinite time, the entropy method shows exponential convergence to equilibrium for renormalised solutions to complex balanced systems with boundary equilibria.


Renormalised solutions Complex balanced reaction networks Reaction–diffusion systems Convergence to equilibrium Entropy method Complex balance equilibria Boundary equilibria 

Mathematics Subject Classification

35B40 35K57 35Q92 80A30 80A32 



Open access funding provided by University of Graz. This work is partially supported by International Research Training Group IGDK 1754 and NAWI Graz. Moreover, the authors would like to acknowledge the referees for the careful reading of the paper and the suggested improvements.


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Authors and Affiliations

  1. 1.Institute of Mathematics and Scientific ComputingUniversity of GrazGrazAustria

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