Stability of traveling waves for autocatalytic reaction systems with strong decay

Yaping Wu; Niannian Yan

doi:10.3934/dcdsb.2017033

Article Contents

2017, Volume 22, Issue 4: 1601-1633. Doi: 10.3934/dcdsb.2017033

This issue Previous Article Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions Next Article Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity

Stability of traveling waves for autocatalytic reaction systems with strong decay

Yaping Wu^, and
Niannian Yan

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

* Corresponding author
* Corresponding author

Received: May 2016

Revised: June 2016

Published: August 2017

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Abstract

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.

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Mathematics Subject Classification: Primary:35C07, 35B35, 35K57, 47A75, 62M15, 34L16.

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Figure 1. Wave profiles by using shooting method for $K=1.5$, $q=2$, $c=3>c^*$ (a) $d=1$, $u^*=1$, (b) $d=3$, $u^*=2.1853$

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Figure 2. 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) $\xi_0\!\!=\!\!-80$

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Figure 3. 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 $sl$

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Figure 4. (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^∗

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Figure 5. 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 $c\!\!>\!\!c^*$

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Figure 6. 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 $c\!\!>\!\!c^*$

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Figure 7. 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 $d\!\in\!(0,d^*)$

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Figure 8. 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 $c\!\!>\!\!c^*$

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Table 1. 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

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