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December  2017, 10(4): 1055-1087. doi: 10.3934/krm.2017042

## The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks

 Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36,8010 Graz, Austria

* Corresponding author: Klemens Fellner

Received  April 2015 Revised  November 2016 Published  March 2017

In this paper, the applicability of the entropy method for the trend towards equilibrium for reaction-diffusion systems arising from first order chemical reaction networks is studied. In particular, we present a suitable entropy structure for weakly reversible reaction networks without detail balance condition.

We show by deriving an entropy-entropy dissipation estimate that for any weakly reversible network each solution trajectory converges exponentially fast to the unique positive equilibrium with computable rates. This convergence is shown to be true even in cases when the diffusion coefficients of all but one species are zero.

For non-weakly reversible networks consisting of source, transmission and target components, it is shown that species belonging to a source or transmission component decay to zero exponentially fast while species belonging to a target component converge to the corresponding positive equilibria, which are determined by the dynamics of the target component and the mass injected from other components. The results of this work, in some sense, complete the picture of trend to equilibrium for first order chemical reaction networks.

Citation: Klemens Fellner, Wolfang Prager, Bao Q. Tang. The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic & Related Models, 2017, 10 (4) : 1055-1087. doi: 10.3934/krm.2017042
##### References:

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##### References:
A first-order chemical reaction network
A reversible network
A non-weakly reversible reaction network consisting of four strongly connected components
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