Acta Applicandae Mathematicae

, Volume 139, Issue 1, pp 25–57 | Cite as

Global Existence for a Class of Reaction-Diffusion Systems with Mass Action Kinetics and Concentration-Dependent Diffusivities

  • Dieter Bothe
  • Guillaume Rolland


In this work, we study the existence of classical solutions for a class of reaction-diffusion systems with quadratic growth naturally arising in mass action chemistry when studying networks of reactions of the type A i +A j A k with Fickian diffusion, where the diffusion coefficients might depend on time, space and on all the concentrations c i of the chemical species. In the case of one single reaction, we prove global existence for space dimensions N≤5. In the more restrictive case of diffusion coefficients of the type d i (c i ), we use an L 2-approach to prove global existence for N≤9. In the general case of networks of reactions we extend the previous method to get global solutions for general diffusivities if N≤3 and for diffusion of type d i (c i ) if N≤5. In the latter quasi-linear case of d i (c i ) and for space dimensions N=2 and N=3, global existence holds for more than quadratic reactions. We can actually allow for more general rate functions including fractional power terms, important in applications. We obtain global existence under appropriate growth restrictions with an explicit dependence on the space dimension N.


Reaction-diffusion systems Reversible reactions Mass action kinetics Global existence Filtration equation 

Mathematics Subject Classification

35K51 35K57 35K59 


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.NantesFrance

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