Existence and Stability of Travelling Front Solutions for General Auto-catalytic Chemical Reaction Systems

¹ Department of Mathematics, Xi’an Jiaotong University, Xi’an 710049, P.R. China and Department of Mathematics and Statistics, Wright State University, Dayton, OH45435, USA
² College of Mathematical Sciences, Capital Normal University, Beijing, 100048, P.R. China

^⋆ Corresponding author. E-mail: yaping_wu@hotmail.com

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(ℝ) 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 L₁(ℝ) norm and decays algebraically in C_unif(ℝ) norm, if the initial perturbation is in addition small in L₁ norm.