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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
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification

35K51 35K57 35K59 

References

  1. 1.
    Amann, H.: Dynamic theory of quasilinear parabolic systems. III. Global existence. Math. Z. 202(2), 219–250 (1989) MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Function Spaces, Differential Operators and Nonlinear Analysis, Friedrichroda, 1992. Teubner-Texte Math., vol. 133, pp. 9–126. Teubner, Stuttgart (1993) CrossRefGoogle Scholar
  3. 3.
    Baras, P.: Compacité de l’opérateur fu solution d’une équation non linéaire (du/dt)+Auf. C. R. Acad. Sci. Paris 268, 1113–1116 (1978) MathSciNetGoogle Scholar
  4. 4.
    Baldyga, J., Bourne, J.R.: Turbulent Mixing and Chemical Reactions. Wiley, New York (1999) Google Scholar
  5. 5.
    Bénilan, Ph., Crandall, M.G., Pazy, A.: Evolution equations in Banach spaces, preprint monograph Google Scholar
  6. 6.
    Bothe, D., Dreyer, W.: Continuum thermodynamics of chemically reacting fluid mixtures and the Maxwell-Stefan equations of multicomponent mass transport (2014), submitted. http://arxiv.org/pdf/1401.5991.pdf
  7. 7.
    Bothe, D.: Flow invariance for perturbed nonlinear evolution equations. Abstr. Appl. Anal. 1, 379–395 (1996) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bothe, D.: Nonlinear evolutions in Banach spaces—Existence and qualitative theory with applications to reaction-diffusion systems. Habilitation thesis, University of Paderborn (1999) Google Scholar
  9. 9.
    Bothe, D.: On the Maxwell-Stefan equations to multicomponent diffusion. In: Progress in Nonlinear Differential Equations and Their Applications, vol. 60, pp. 81–93. Springer, Basel (2011) Google Scholar
  10. 10.
    Bothe, D., Pierre, M.: Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate. J. Math. Anal. Appl. 368, 120–132 (2010) MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Bothe, D., Pierre, M., Rolland, G.: Cross-diffusion limit for a reaction-diffusion system with fast reversible reaction. Commun. Partial Differ. Equ. 37, 1940–1966 (2012) MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Choi, Y.S., Lui, R., Yamada, Y.: Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion. Discrete Contin. Dyn. Syst. 9, 1193–1200 (2003) MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Cussler, E.L.: Diffusion: Mass Transfer in Fluid Systems, 2nd edn. Cambridge University Press, Cambridge (1997) Google Scholar
  14. 14.
    Desvillettes, L., Fellner, K.: Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations. J. Math. Anal. Appl. 319, 157–176 (2006) MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    de Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. Dover, New York (1984) Google Scholar
  16. 16.
    Denk, R., Hieber, M., Prüss, J.: \(\mathcal {R}\)-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic type. Mem. Am. Math. Soc. 166(788), 1–114 (2003) Google Scholar
  17. 17.
    Desvillettes, L., Lepoutre, Th., Moussa, A.: Entropy, duality, and cross diffusion. SIAM J. Math. Anal. 46, 820–853 (2014) MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Dung, L.: Global existence for a class of strongly coupled parabolic systems. Ann. Mat. Pura Appl. (4) (2006) Google Scholar
  19. 19.
    Espenson, J.H.: Chemical Kinetics and Reaction Mechanisms, 2nd edn. McGraw-Hill, New York (1995) Google Scholar
  20. 20.
    Érdi, P., Tóth, J.: Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models. Nonlinear Science: Theory and Applications. Princeton University Press, Princeton (1989) MATHGoogle Scholar
  21. 21.
    Feng, W.: Coupled system of reaction-diffusion equations and applications in carrier facilitated diffusion. Nonlinear Anal. 17(3), 285–311 (1991) MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Giovangigli, V.: Multicomponent Flow Modeling. Birkhäuser, Boston (1999) MATHCrossRefGoogle Scholar
  23. 23.
    Goudon, T., Vasseur, A.: Regularity analysis for systems of reaction-diffusion equations. Ann. Sci. Éc. Norm. Super. 43, 117–142 (2010) MATHMathSciNetGoogle Scholar
  24. 24.
    Ito, K., Kappel, F.: Evolution Equations and Approximation. World Scientific, Singapore (2002) Google Scholar
  25. 25.
    Jüngel, A., Stelzer, I.V.: Existence analysis of Maxwell-Stefan systems for multicomponent mixtures. SIAM J. Math. Anal. (2013) Google Scholar
  26. 26.
    Krishna, R., Taylor, R.: Multicomponent Mass Transfer. Wiley, New York (1993) Google Scholar
  27. 27.
    Lichtner, P.C., Steefel, C.I., Oelkers, E.H.: Reactive Transport in Porous Media. Reviews in Mineralogy, vol. 34. Mineralogical Society of America, Washington (1996) Google Scholar
  28. 28.
    Ladyzenskaja, O.A., Solonnikov, V.A., Ural’seca, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translation of Mathematical Monographs, vol. 23. Am. Math. Soc., Providence (1968) Google Scholar
  29. 29.
    Murray, J.D.: Mathematical Biology. Springer, Berlin (1989) MATHCrossRefGoogle Scholar
  30. 30.
    Morgan, J., Waggonner, S.: Global existence for a class of quasilinear reaction-diffusion systems. Commun. Appl. Anal. 8, 153–166 (2004) MATHMathSciNetGoogle Scholar
  31. 31.
    Pierre, M.: Global existence in reaction-diffusion systems with control of mass: a survey. Milan J. Math. 78, 417–455 (2010) MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Prüss, J.: Maximal regularity for evolution equations in L p-spaces. Conf. Semin. Mat. Univ. Bari 285, 1–39 (2003) Google Scholar
  33. 33.
    Rothe, F.: Global Solutions of Reaction-Diffusion Systems. Lecture Notes in Mathematics, vol. 1072. Springer, Berlin (1984) MATHGoogle Scholar
  34. 34.
    Szabo, Z.G.: Advances in Kinetics of Homogeneous Gas Reactions. Methuen, London (1964) Google Scholar
  35. 35.
    Vazquez, J.L.: The Porous Medium Equation—Mathematical Theory. Clarendon, Oxford (2007) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.NantesFrance

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