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Spatiotemporal patterns in the Lengyel-Epstein reaction-diffusion model
- Yaying Dong^{1},
- Shunli Zhang^{1} and
- Shanbing Li^{2}Email author
https://doi.org/10.1186/s13662-016-0757-y
© Dong et al. 2016
- Received: 13 October 2015
- Accepted: 13 January 2016
- Published: 26 January 2016
Abstract
In this paper, we continue the mathematical study started in (Jang et al. in J. Dyn. Differ. Equ. 16:297-320, 2004; Ni and Tang in Trans. Am. Math. Soc. 357:3953-3969, 2005) on the analytic aspects of the Lengyel-Epstein reaction-diffusion system. First, we further analyze the fundamental properties of nonconstant positive solutions. On the other hand, we continue to consider the effect of the diffusion coefficient d. We obtain another nonexistence result for the case of large d by the implicit function theory, and investigate the direction of bifurcation solutions from \((u^{*},v^{*})\). These results promote the Turing patterns arising from the Lengyel-Epstein reaction-diffusion system.
Keywords
- Lengyel-Epstein system
- nonexistence
- implicit function theory
- bifurcation direction
- fundamental properties
1 Introduction
In the past decade, the Lengyel-Epstein reaction-diffusion system (1.1) and (1.2) have been extensively studied by several authors. For example, the authors gave various important experimental and numerical studies in [8, 9] and the references therein. In the one-dimensional case, Yi et al. [10], regarding b as the bifurcation parameter, studied the Hopf bifurcation for both the ODE and the PDE models. In [11], they further investigated the global asymptotical behavior of constant positive solution for small a. Taking b as the bifurcation parameter, Du and Wang [12] gave the existence of multiple spatially non-homogeneous periodic solutions though all the parameters of the system are spatially homogeneous. Furthermore, by choosing the different bifurcation parameter, Jin et al. [13] considered the same model using similar methods to [12].
On the other hand, Ni and Tang [1, 2] proposed more original and systematic works on mathematical aspects. In [2], for the better description of the structures, they considered the global bifurcation of the nonconstant steady states emanating from the simple bifurcation (i.e., the case \(d_{j}\neq d_{k}\)) by choosing d as bifurcation parameter. In [1], they reported some fundamental analytic properties, and investigated the nonexistence of Turing patterns and the Turing instability. Moreover, they showed that if the parameter a lies in a suitable range, then the system (1.2) possesses nonconstant steady states for large d. In [14], the authors still viewed the effective diffusion rate d as the bifurcation parameter, and maintained the basic hypothesis on the system parameters. They studied the Turing structures, especially bifurcating from the double eigenvalue (i.e., the case \(d_{j} = d_{k}\)) by using the Lyapunov-Schmidt technique and singularity theory [15], and they further discussed the stability and multiplicity of the bifurcating solutions.
In the present paper, based on the results of Ni et al., we continue the analytic works of [1, 2] with the goal of achieving a deeper understanding of the Turing patterns operating in the system (1.2). This paper is organized as follows. In Section 2, by the implicit function theory, we consider the nonexistence result for the case of large d. In Section 3, we continue to analyze the fundamental properties of nonconstant positive solutions. These two sections complete the work of [1]. Finally, in Section 4, we investigate the direction of bifurcation solutions from simple eigenvalue (i.e., the case \(d_{j}\neq d_{k}\)), which promotes the results in [2].
2 The nonexistence of nonconstant steady states
In this section, we shall verify the nonexistence of nonconstant steady states for the case of large d. To this end, we recall some results in [1]. First, we state a priori estimates of upper and lower bounds for positive solutions to the problem (1.2).
Lemma 2.1
[1]
The following theorem gives the nonexistence of nonconstant steady states to the problem (1.2) when d is not large.
Theorem 2.1
[1]
We remark that it is very involved to derive a good estimate for the positive constant \(d_{0}\) obtained above for a given a. However, Ni and Tang obtained a much simpler estimate in a different way when a is not very large.
Theorem 2.2
[1]
Theorem 2.1 and Theorem 2.2 give the nonexistence results for the case of not large d. In the following, we continue to analyze the effect of the parameter d on the nonexistence of nonconstant steady states to the problem (1.2). To obtain the nonexistence result for the case of large d, we first give the asymptotic behavior of positive solutions to (1.2) when d is sufficiently large.
Lemma 2.2
Suppose that \((u,v)=(u(x),v(x))\) is any positive solution of (1.2). Then \((u,v)\rightarrow(u^{*},v^{*})\) in \([C^{2}(\overline{\Omega})]^{2}\) as \(d\rightarrow\infty\).
Proof
Suppose that a and Ω are fixed. By Lemma 2.1 and the standard elliptic regularity theory, we may assume that for any positive solution sequence \((u, v)\) of (1.2) with respect to d, there exists a subsequence \(\{(u_{i} , v_{i})\}_{i=1}^{\infty}\) corresponding to \(d = d_{i}\) with \(d_{i}\rightarrow\infty\) as \(i\rightarrow\infty\), such that \((u_{i} , v_{i})\rightarrow(\tilde{u},\tilde{v})\) in \([C^{2}(\overline{\Omega})]^{2}\) as \(i\rightarrow\infty\).
Now, by Lemma 2.2, we can apply the implicit function theorem to obtain the nonexistence result for the case of large d.
Theorem 2.3
If \(a^{2}< 125/3\), then there exists a large constant \(d^{*}=d^{*}(a,\Omega)\), such that the problem (1.2) does not admit a nonconstant solution provided that \(d>d^{*}\).
Proof
By virtue of the implicit function theorem, we see that there exist two positive constants \(\rho_{0} \) and \(r_{0}\), which are sufficiently small, such that for any \(\rho\in(0,\rho_{0}]\), \((u^{*}, 0, v^{*})\) is the unique solution of \({F}(\rho, u, v_{1}, v_{2}) = 0\) in \(B_{r_{0}} (u^{*}, 0, v^{*})\), where \(B_{r_{0}} (u^{*}, 0, v^{*})\) denotes the open ball in \(W^{2,2}_{n}(\Omega)\times(W^{2,2}_{n}(\Omega)\cap L^{2}_{0}(\Omega ))\times R_{+}^{1}\) centered at \((u^{*}, 0, v^{*})\) with radius \(r_{0}\).
3 Properties of nonconstant positive solutions
In the section, based on the results of Ni and Tang [1], we continue to investigate the basic properties of nonconstant positive solutions to the Lengyel-Epstein reaction-diffusion system.
Lemma 3.1
Proof
Lemma 3.2
Proof
By (3.1) and (3.2), we have the following theorem.
Theorem 3.1
Next, we promote the relationship of the gradients of u and v based on the work of Ni and Tang. It is to be found that our proof does not depend on the previous estimates. For this purpose, we need to introduce some results.
Lemma 3.3
[1]
Lemma 3.4
[1]
Now, by Lemma 3.3 and Lemma 3.4, our purpose is to obtain a new result on the relationship of the gradients of u and v.
Theorem 3.2
Proof
4 Direction of the bifurcation solutions
From the above analysis, we obtain the following results.
5 Conclusion
In this paper, we have studied the Lengyel-Epstein reaction-diffusion system which is proposed by Lengyel and Epstein in [3, 4]. Based on the results of Ni et al. [1, 2], we further study the steady-state problem (1.2). By the implicit function theory, we have shown that if the feed concentration is not large (\(a^{2}< 125/3\)), then the chemical concentrations of iodide (\(\mathrm{I}^{-}\)) and chlorite (\(\mathrm{ClO}_{2}^{-}\)) remain unchanged (i.e., \((u^{*},v^{*})\)) when d is sufficiently large (see Theorem 2.3). For the chemical concentrations of iodide (\(\mathrm{I}^{-}\)) and chlorite (\(\mathrm{ClO}_{2}^{-}\)), we have obtained better estimates (see Theorem 3.1 and Theorem 3.2). Furthermore, by using the work of Shi [17], we have determined the change of the chemical concentrations of iodide (\(\mathrm{I}^{-}\)) and chlorite (\(\mathrm{ClO}_{2}^{-}\)) close to \((u^{*},v^{*})\) (see Theorem 4.1).
Declarations
Acknowledgements
The work is supported by the Natural Science Foundation of China (11371293). The authors would like to express their sincere thanks to the anonymous referees for their valuable suggestions which led to the improved presentation of the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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