WIAS Preprint No. 3008, (2023)

Convergence to self-similar profiles in reaction-diffusion systems



Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Schindler, Stefanie
    ORCID: 0000-0002-4005-7314

2020 Mathematics Subject Classification

  • 35K57 35C06 35B45

Keywords

  • Mass-action kinetics, relative Boltzmann entropy, infinite-mass systems, energy-dissipation estimates, self-similar profiles

DOI

10.20347/WIAS.PREPRINT.3008

Abstract

We study a reaction-diffusion system on the real line, where the reactions of the species are given by one reversible reaction pair satisfying the mass-action law. We describe different positive limits at both sides of infinityand investigate the long-time behavior. Rescaling space and time according to the parabolic scaling, we show that solutions converge exponentially to a similarity profile when the scaled time goes to infinity. In the original variables, these profiles correspond to asymptotically self-similar behavior describing the phenomenon of diffusive mixing of the different states at infinity.Our method provides global exponential convergence for all initial states with finite relative entropy. For the case with equal stoichiometric coefficients, we can allow for self-similar profiles with arbitrary equilibrated states,while in the other case we need to assume that the two states atinfinity are sufficiently close such that the self-similar profile is relative flat.

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