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June  2017, 22(4): 1601-1633. doi: 10.3934/dcdsb.2017033

## Stability of traveling waves for autocatalytic reaction systems with strong decay

 School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

* Corresponding author

Received  May 2016 Revised  June 2016 Published  November 2016

This paper is concerned with the spatial decay and stability of travelling wave solutions for a reaction diffusion system $u_{t}=d u_{xx}-uv$, $v_{t}=v_{xx}+uv-Kv^{q}$ with $q>1$. By applying Centre Manifold Theorem and detailed asymptotic analysis, we get the precise spatial decaying rate of the travelling waves with noncritical speeds. Further by applying spectral analysis, Evans function method and some numerical simulation, we proved the spectral stability and the linear exponential stability of the waves with noncritical speeds in some weighted spaces.

Citation: Yaping Wu, Niannian Yan. Stability of traveling waves for autocatalytic reaction systems with strong decay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1601-1633. doi: 10.3934/dcdsb.2017033
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##### References:
Wave profiles by using shooting method for $K=1.5$, $q=2$, $c=3>c^*$ (a) $d=1$, $u^*=1$, (b) $d=3$, $u^*=2.1853$
Wave profiles for $d\!\!=\!\!1$, $u^*\!\!=\!\!1$, $c\!\!=\!\!3\!\!>\!\!c^*$, $K\!=\!1.5$, $q\!\!=\!\!2$, $sl\!\!=\!\!-500$, $sr\!\!=\!\!50$ and (a) $\xi_0\!\!=\!\!0$; (b) $\xi_0\!\!=\!\!-80$
Wave profiles for $K\!\!=\!\!1.5$, $d\!\!=\!\!1$, $u^*\!\!=\!\!1$, $q\!=\!2$, $sr\!\!=\!\!50$, $\xi_0\!=\!0$ and (a) $sl\!\!=\!\!-500$ with different $c\!\!>\!\!c^*$; (b) $c\!\!=\!\!4$ with different starting point $sl$
(a) The selected curve Γ for K=1:5, q=2, u=1, c=3 and d=1. (b) The numerical curves of E(Γ) for q = 1:5, K = 1:5 and d = 1 with different c > c
The numerical curves of $E(\Gamma)$ for $q\!=\!2$ and $K\!\!>\!\!1$: (a) $K\!\!=\!\!1.5$, $c\!\!=\!\!3$ with different $d\!\in\!(0,d^*)$, (b) $d\!\!=\!\!1$, $K\!\!=\!\!1.5$ with different $c\!\!>\!\!c^*$
The numerical curves of $E(\Gamma)$ for $q\!=\!2$ and $0\!\!<\!\!K\!\!\leq\!\!1$: (a) $K\!\!=\!\!0.5$, $c\!\!=\!\!3$ with different $d\!\in\!(0,d^*)$, (b) $d\!\!=\!\!1$, $K\!\!=\!\!0.5$ with different $c\!\!>\!\!c^*$
The numerical curves of $E(\Gamma)$ for $1\!<q\!<2$: (a) $d\!\!=\!\!1$, $K\!\!=\!\!1.5$, $c\!\!=\!\!4$ with different $q\!\!\in\!\!(1,2)$, (b) $q\!\!=\!\!1.5$, $K\!=\!1.5$, $c\!\!=\!\!4$ with different $d\!\in\!(0,d^*)$
The numerical curves of $E(\Gamma)$ for $q=2.5$: (a) $K\!\!=\!\!1.5$, $c\!\!=\!\!4$ with different $d\!\in\!(0,d^*)$ (b) $d\!\!=\!\!1$, $K\!\!=\!\!1.5$ with different $c\!\!>\!\!c^*$
The values of $E(0)$ and $E(10^4)$ for $q=2$, $K>1$ with the parameters corresponding to the curves in Fig. 5, we just retain integers here, which is verified the estimates in Lemma 4.5
 $K\!=\!1.5$, $c\!=\!3$ $c\!=\!3$, $d\!=\!1$ $K\!=\!1.5$, $d\!=\!1$ $d\!=\!0.7$ $d\!=\!1$ $d\!=\!3$ $K\!=\!5$ $K\!=\!3$ $K\!=\!1.5$ $c\!=\!3$ $c\!=\!5$ $c\!=\!10$ $E(0)$ 109 77 23 177 92 68 78 66 136 $E(10^4)$ 49254 41166 23672 41633 41262 41093 41166 40480 40733
 $K\!=\!1.5$, $c\!=\!3$ $c\!=\!3$, $d\!=\!1$ $K\!=\!1.5$, $d\!=\!1$ $d\!=\!0.7$ $d\!=\!1$ $d\!=\!3$ $K\!=\!5$ $K\!=\!3$ $K\!=\!1.5$ $c\!=\!3$ $c\!=\!5$ $c\!=\!10$ $E(0)$ 109 77 23 177 92 68 78 66 136 $E(10^4)$ 49254 41166 23672 41633 41262 41093 41166 40480 40733
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