Existence and uniqueness of solution of a continuous flow bioreactor model with two species

  • M. CrespoEmail author
  • B. Ivorra
  • A. M. Ramos
Original Paper


In this work, we perform the mathematical analysis of a coupled system of two reaction–diffusion–advection equations and Danckwerts boundary conditions, which models the interaction between a microbial population (e.g., bacterias) and a diluted substrate (e.g., nitrate) in a continuous flow bioreactor. This type of bioreactor can be used, for instance, for water treatment. First, we prove the existence and uniqueness of solution, under the hypothesis of linear reaction by using classical results for linear parabolic boundary value problems. Next, we prove the existence and uniqueness of solution for some nonlinear reactions by applying Schauder Fixed Point Theorem and the theorem obtained for the linear case. Results about the nonnegativeness and boundedness of the solution are also proved here.


Existence Uniqueness Reaction–diffusion–advection Nonlinear parabolic system Bioreactor 



This work was carried out thanks to the financial support of the Spanish “Ministry of Economy and Competitiveness” under project MTM2011-22658; the research group MOMAT (Ref. 910480) supported by “Banco Santander” and “Universidad Complutense de Madrid”; and the “Junta de Andalucía” and the European Regional Development Fund through project P12-TIC301.


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Copyright information

© Springer-Verlag Italia 2015

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada, Instituto de Matemática InterisciplinarUniversidad Complutense de MadridMadridSpain

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