### Abstract

This paper is concerned with the existence and stability of travelling front solutions for more general autocatalytic chemical reaction systems u _{t} = du_{xx}-uf(v), v_{t} = v_{xx} + 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 C_{unif} 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 C_{unif} norm, if the initial perturbation is in addition small in L_{1} norm.

Original language | English |
---|---|

Pages (from-to) | 2-17 |

Number of pages | 16 |

Journal | Mathematical Modelling of Natural Phenomena |

Volume | 8 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2013 Jun 28 |

### Fingerprint

### Keywords

- Existence
- Spectral analysis
- Stability
- Travelling waves

### ASJC Scopus subject areas

- Modelling and Simulation
- Applied Mathematics

### Cite this

*Mathematical Modelling of Natural Phenomena*,

*8*(3), 2-17. https://doi.org/10.1051/mmnp/20138308

**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.

Research output: Contribution to journal › Article

*Mathematical Modelling of Natural Phenomena*, vol. 8, no. 3, pp. 2-17. https://doi.org/10.1051/mmnp/20138308

}

TY - JOUR

T1 - Existence and stability of travelling front solutions for general auto-catalytic chemical reaction systems

AU - Crooks, Elaine

AU - Davidson, Fordyce

AU - Kazmierczak, Bogdan

AU - Nadin, Gregoire

AU - Tsai, Je-Chiang

AU - Li, Y.

AU - Wu, Y.

PY - 2013/6/28

Y1 - 2013/6/28

N2 - 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.

AB - 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.

KW - Existence

KW - Spectral analysis

KW - Stability

KW - Travelling waves

UR - http://www.scopus.com/inward/record.url?scp=84879292490&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84879292490&partnerID=8YFLogxK

U2 - 10.1051/mmnp/20138308

DO - 10.1051/mmnp/20138308

M3 - Article

AN - SCOPUS:84879292490

VL - 8

SP - 2

EP - 17

JO - Mathematical Modelling of Natural Phenomena

JF - Mathematical Modelling of Natural Phenomena

SN - 0973-5348

IS - 3

ER -