Advertisement

Journal of Evolution Equations

, Volume 18, Issue 4, pp 1713–1720 | Cite as

Global existence for reaction–diffusion systems with dissipation of mass and quadratic growth

  • Philippe Souplet
Article
  • 55 Downloads

Abstract

We consider the Neumann and Cauchy problems for positivity preserving reaction–diffusion systems of m equations enjoying the mass and entropy dissipation properties. We show global classical existence in any space dimension, under the assumption that the nonlinearities have at most quadratic growth. This extends previously known results which, in dimensions \(n\ge 3\), required mass conservation and were restricted to the Cauchy problem. Our proof is also simpler.

Keywords

Reaction–diffusion systems Mass dissipation Entropy Global existence 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Bisi, L. Desvillettes and G. Spiga, Exponential convergence to equilibrium via Lyapounov functionals for reaction–diffusion equations arising from non reversible chemical kinetics, ESAIM Math. Model. Numer. Anal. 43 (2009), 151–172.MathSciNetCrossRefGoogle Scholar
  2. 2.
    J.A. Cañizo, L. Desvillettes and K. Fellner, Improved duality estimates and applications to reaction–diffusion equations, Comm. Partial Differential Equations 39 (2014), 1185–1204.MathSciNetCrossRefGoogle Scholar
  3. 3.
    M.C. Caputo, T. Goudon, A. Vasseur, Solutions of the \(4\)-species quadratic reaction–diffusion system are bounded and \(C^\infty \)-smooth, in any space dimension, Preprint arXiv:1709.05694 (2017).
  4. 4.
    M.C. Caputo and A. Vasseur, Global regularity of solutions to systems of reaction–diffusion with sub-quadratic growth in any dimension, Comm. Partial Differential Equations 34 (2009), 1228–1250.MathSciNetCrossRefGoogle Scholar
  5. 5.
    L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl. 319 (2006), 157–176.MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, Global existence for quadratic systems of reaction–diffusion, Adv. Nonlinear Stud. 7 (2007), 491–511.MathSciNetCrossRefGoogle Scholar
  7. 7.
    K. Fellner and E.-H. Laamri, Exponential decay towards equilibrium and global classical solutions for nonlinear reaction–diffusion systems, J. Evol. Equ. 16 (2016), 681–704.MathSciNetCrossRefGoogle Scholar
  8. 8.
    K. Fellner, W. Prager and B.Q. Tang, Exponential decay towards equilibrium and global classical solutions for nonlinear reaction–diffusion systems, Kinet. Relat. Models 10 (2017), 1055–1087.MathSciNetCrossRefGoogle Scholar
  9. 9.
    K. Fellner and B.Q. Tang, Explicit exponential convergence to equilibrium for nonlinear reaction–diffusion systems with detailed balance condition, Nonlinear Anal. 159 (2017), 145–180.MathSciNetCrossRefGoogle Scholar
  10. 10.
    M. Fila and H. Ninomiya, Reaction versus diffusion: blow-up induced and inhibited by diffusivity, Russian Math. Surveys 60 (2005), 1217–1235.MathSciNetCrossRefGoogle Scholar
  11. 11.
    T. Goudon and A. Vasseur, Regularity analysis for systems of reaction-diffusion equations, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 117–142.Google Scholar
  12. 12.
    J.I. Kanel, The Cauchy problem for a system of semilinear parabolic equations with balance conditions, Differentsial’nye Uravneniya 20 (1984), 1753–1760 (English translation: Differential Equations 20 (1984), 1260–1266).Google Scholar
  13. 13.
    J.I. Kanel, Solvability in the large of a system of reaction–diffusion equations with the balance condition, Differentsial’nye Uravneniya 26 (1990), 448–458 (English translation: Differential Equations 26 (1990), 331–339.Google Scholar
  14. 14.
    A. Mielke, J. Haskovec and P.A. Markowich, On uniform decay of the entropy for reaction–diffusion systems, J. Dynam. Differential Equations 27 (2015), 897–928.MathSciNetCrossRefGoogle Scholar
  15. 15.
    X. Mora, Semilinear parabolic equations define semiflows on \(C^k\) spaces, Trans. Amer. Math. Soc. 278 (1983), 21–55.MathSciNetzbMATHGoogle Scholar
  16. 16.
    M. Pierre, Global existence in reaction–diffusion systems with control of mass: a survey, Milan J. Math. 78 (2010), 417–455.MathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Pierre, T. Suzuki and Y. Yamada, Dissipative reaction diffusion systems with quadratic growth, Indiana Univ. Math. J. (2018), to appear (Preprint hal: 01671797).Google Scholar
  18. 18.
    M. Pierre, T. Suzuki and R. Zou, Asymptotic behavior of solutions to chemical reaction–diffusion systems, J. Math. Anal. Appl. 450 (2017), 152–168.MathSciNetCrossRefGoogle Scholar
  19. 19.
    P. Quittner, Ph. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states, Birkhäuser Advanced Texts, 2007, 584 p.+xi.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université Paris 13, Sorbonne Paris Cité, CNRS UMR 7539, Laboratoire Analyse Géométrie et ApplicationsVilletaneuseFrance

Personalised recommendations