The entropy dissipation method for spatially inhomogeneous reaction-diffusion-type systems

Marco Di Francesco, Klemens Fellner, Peter A. Markowich

Research output: Contribution to journalArticlepeer-review

23 Scopus citations


We study the long-time asymptotics of reaction-diffusion-type systems that feature a monotone decaying entropy (Lyapunov, free energy) functional. We consider both bounded domains and confining potentials on the whole space for arbitrary space dimensions. Our aim is to derive quantitative expressions for (or estimates of) the rates of convergence towards an (entropy minimizing) equilibrium state in terms of the constants of diffusion and reaction and with respect to conserved quantities. Our method, the so-called entropy approach, seeks to quantify convergence to equilibrium by using functional inequalities, which relate quantitatively the entropy and its dissipation in time. The entropy approach is well suited to nonlinear problems and known to be quite robust with respect to model variations. It has already been widely applied to scalar diffusion-convection equations, and the main goal of this paper is to study its generalization to systems of partial differential equations that contain diffusion and reaction terms and admit fewer conservation laws than the size of the system. In particular, we successfully apply the entropy approach to general linear systems and to a nonlinear example of a reaction-diffusion-convection system arising in solid-state physics as a paradigm for general nonlinear systems.

Original languageEnglish (US)
Pages (from-to)3273-3300
Number of pages28
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2100
StatePublished - Dec 8 2008
Externally publishedYes


  • Entropy-entropy dissipation approach
  • Long-time asymptotics
  • Reaction-diffusion systems

ASJC Scopus subject areas

  • General Engineering
  • General Physics and Astronomy
  • General Mathematics


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