Existence and stability of travelling front solutions for general auto-catalytic chemical reaction systems

Elaine Crooks, Fordyce Davidson, Bogdan Kazmierczak, Gregoire Nadin, Je-Chiang Tsai, Y. Li, Y. Wu

Research output: Contribution to journalArticle

Abstract

This paper is concerned with the existence and stability of travelling front solutions for more general autocatalytic chemical reaction systems u t = duxx-uf(v), vt = vxx + uf(v) with d > 0 and d ≠ 1, where f(v) has super-linear or linear degeneracy at v = 0. By applying Lyapunov-Schmidt decomposition method in some appropriate exponentially weighted spaces, we obtain the existence and continuous dependence of wave fronts with some critical speeds and with exponential spatial decay for d near 1. By applying special phase plane analysis and approximate center manifold theorem, the existence of traveling waves with algebraic spatial decay or with some lower exponential decay is also obtained for d > 0. Further, by spectral estimates and Evans function method, the wave fronts with exponential spatial decay are proved to be spectrally or linearly stable in some suitable exponentially weighted spaces. Finally, by adopting the main idea of proof in [12] and some similar arguments as in [21], the waves with critical speeds or with non-critical speeds are proved to be locally exponentially stable in some exponentially weighted spaces and Lyapunov stable in Cunif space, if the initial perturbation of the waves is small in both the weighted and unweighted norms; the perturbation of the waves also stays small in L 1 norm and decays algebraically in Cunif norm, if the initial perturbation is in addition small in L1 norm.

Original languageEnglish
Pages (from-to)2-17
Number of pages16
JournalMathematical Modelling of Natural Phenomena
Volume8
Issue number3
DOIs
Publication statusPublished - 2013 Jun 28

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Travelling Fronts
Chemical Reaction
Chemical reactions
Weighted Spaces
Decay
Perturbation
Norm
Wave Front
Lyapunov-Schmidt Method
Phase Plane Analysis
Evans Function
Center Manifold Theorem
L1-norm
Continuous Dependence
Exponential Decay
Decomposition Method
Degeneracy
Traveling Wave
Lyapunov
Linearly

Keywords

  • Existence
  • Spectral analysis
  • Stability
  • Travelling waves

ASJC Scopus subject areas

  • Modelling and Simulation
  • Applied Mathematics

Cite this

Existence and stability of travelling front solutions for general auto-catalytic chemical reaction systems. / Crooks, Elaine; Davidson, Fordyce; Kazmierczak, Bogdan; Nadin, Gregoire; Tsai, Je-Chiang; Li, Y.; Wu, Y.

In: Mathematical Modelling of Natural Phenomena, Vol. 8, No. 3, 28.06.2013, p. 2-17.

Research output: Contribution to journalArticle

Crooks, Elaine ; Davidson, Fordyce ; Kazmierczak, Bogdan ; Nadin, Gregoire ; Tsai, Je-Chiang ; Li, Y. ; Wu, Y. / Existence and stability of travelling front solutions for general auto-catalytic chemical reaction systems. In: Mathematical Modelling of Natural Phenomena. 2013 ; Vol. 8, No. 3. pp. 2-17.
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