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Stability analysis on steady-state bifurcation for arbitrary order autocatalytic reaction model
Accepted Manuscript Stability analysis on steady-state bifurcation for arbitrary order autocatalytic reaction model
Jiantang Zhao, Yunfeng Jia
PII: DOI: Reference:
S0893-9659(19)30128-4 https://doi.org/10.1016/j.aml.2019.03.027 AML 5840
To appear in:
Applied Mathematics Letters
Received date : 30 November 2018 Revised date : 20 March 2019 Accepted date : 20 March 2019 Please cite this article as: J. Zhao and Y. Jia, Stability analysis on steady-state bifurcation for arbitrary order autocatalytic reaction model, Applied Mathematics Letters (2019), https://doi.org/10.1016/j.aml.2019.03.027 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Stability analysis on steady-state bifurcation for arbitrary order autocatalytic reaction model ∗ Jiantang Zhaoa , Yunfeng Jiab† a College of Mathematics and Information Science, Xianyang Normal University, Xianyang, Shaanxi 712000, China b School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710062, China
Abstract This paper deals with an autocatalytic reaction-diffusion model with arbitrary order functional response subject to no-flux boundary conditions. We mainly discuss the stability of the steady-state bifurcation which emanates from the unique positive constant equilibrium. On the stability of the bifurcation solution, the conventional way is to consider the sign of the first derivative of a certain function. However, sometimes, the first derivative may be equal to zero. This leads to the uncertainty of the stability. In such case, it needs to break through the common idea. We present an approach which determines the stability of the bifurcation solution. Keywords Autocatalytic reaction model; Steady-state bifurcation; Stability. Classifications 35J55, 35K57, 35Q80.
1
Introduction
In numerous chemical processes, when the reactions are far away from the thermodynamic equilibrium, the resemble behaviors between dynamical instability and the equilibrium phase changing on the steady-states in certain systems may likely occur. These behaviors are usually very important dynamical properties of chemical reactions and have been paid much attentions in recent decades. By analyzing these dynamics mathematically, people find that the reactions emerge tremendous fluctuation near the critical points, and then the chemical oscillation occurs as the reaction going on, which may cause the reactions to slow down, and then the system may give rise to critical transition to pattern formation when the phases are changed [1-2]. To study the chemical oscillation in chemical reaction-diffusion systems, it is important to master the kinetic generalities of these ∗
The work is supported in part by the Natural Science Foundations of China (11771262, 61672021), and by the Natural Science Basic Research Plan in Shaanxi Province of China (2018JQ1020). † Corresponding and co-first author.
1
systems near the critical points. These are the main concerns of autocatalytic reaction models, which have been applied to various problems in chemistry and biology, see [3-6] and the references therein for example. The reaction mechanism of the simplest autocatalytic reaction can be given by A + B → 2B, where A and B represent the reactant and autocatalyst, respectively. This reaction is one in which a molecule of species A interacts with a molecule of species B. An A molecule is converted into a B molecule. The final product consists of the original B molecule plusing the B molecule created in the reaction. A more complicated autocatalytic reaction is given by P → A,
A + 2B → 3B,
B → C,
where P is the density of the reaction precursor, A and B also represent the reactant and autocatalyst, C is some certain inert product. The supply of reactant is proceeded by the gradually degenerative process of the precursor. It is assumed that the reactant is effectively immobilised within the reactor, and the autocatalyst is made to flow through the reactor with a constant velocity as well as being able to diffuse, see [7] for details. Corresponding to this reaction mechanism, much of the previous work is concerned with − the iodate-arsenous acid systems, specifically, A and B represent IO− 3 and I , respectively [8]. The basic idea of this type reaction is that the diffusion of the autocatalyst has a stabilizing effect on the planar waves, whereas the diffusion of the substrate has a destabilizing effect. More complex autocatalytic reactions are involved in metabolic pathways and metabolic networks in biological systems. In this paper, we consider the simplified and non-dimensional form of arbitrary order autocatalytic model ut − d1 ∆u = a − uv p , x ∈ Ω, t > 0, p vt − d2 ∆v = uv − v, x ∈ Ω, t > 0, (1.1) ∂u ∂v = = 0, x ∈ ∂Ω, t > 0, ∂ν ∂ν u = u0 ≥, ̸≡ 0, v = v0 ≥, ̸≡ 0, x ∈ Ω, t = 0,
where Ω ⊂ Rn is bounded with smooth boundary ∂Ω, the variables u and v represent the dimensionless concentrations or densities of the reactant and autocatalyst respectively and so are usually assumed to be non-negative. d1 and d2 are the diffusion rates of u and v, respectively, a is a parameter representing the initial concentration of the reaction precursor, p ≥ 2 denotes the order of the reaction with respect to the autocatalytic species, ∂ a, d1 and d2 are positive constants. ∂ν denotes the differential in the direction of the outer normal to ∂Ω. The chemical reaction is driven within a closed container. For the derivation of model (1.1), we refer the readers to [9-10] and references therein for a more detailed description. In [11], Guo et al. considered the steady-state problem corresponding to model (1.1) in one dimensional case, specifically, Ω = (0, π). They mainly investigated the existence of Hopf bifurcation and the local steady-state bifurcation emitting from simple and double
2
eigenvalues. Then in [12], Jia et al. discussed the local steady-state bifurcation emitting from the unique positive constant equilibrium in high dimensional case and for p = 2. However, the stability of the steady-state bifurcation solution was not involved in both references. With what in mind, in this paper, we mainly deal with the stability of the local bifurcation solution bifurcating from the unique positive constant equilibrium U ∗ = (a1−p , a)T to the steady-state problem corresponding (1.1), since the stability of the steadystates is a very important property of dynamical systems which determines the progress of chemical reactions or the population evolution in a large extent (actually, there have been a large number of literatures focusing on the stability of different kinds of dynamical systems, such as [1-2, 5, 8, 12-16], etc). We use the conventional approach introduced in [17] when we consider the sign of the first derivative of a certain function. However, it should be pointed out that an idea is also presented when there is trouble in using the first derivative, which is analyzed and clarified after the proof of the main theorem.
2
Main results
Let µi , i = 0, 1, 2, · · · be all eigenvalues of the operator −∆ with homogenous Neumann boundary conditions. By using the very similar techniques as in proving Theorem 6.1 in [12], and combining with Theorem 3.1 (i) in [11], the following result can be easily obtained, and so we omit the proof here. Theorem 2.1. Assume that there is i > 0 satisfying d2 µi < p − 1. Set ai =
(
d1 µi (p − 1 − d2 µi ) 1 + d2 µi
)1
p
,
i = 1, 2, · · · .
Suppose that the followings hold. (i) ai ̸= aj for any integer i ̸= j; (ii) a = ai0 for some i0 ; (iii) µi0 is simple. Then (Ui∗0 ; ai0 ) is a bifurcation point to the steady-state problem of (1.1), where Ui∗0 = T (a1−p i0 , ai0 ) is the unique positive constant equilibrium to (1.1). Following Theorem 2.1 and according to the Crandall-Rabinowitz bifurcation theorem [18], we know that there exist δ > 0 and smooth functions β : (−δ, δ) −→ R, (ω1 , ω2 )T : (−δ, δ) −→ X satisfying β(0) = 0, ω1 (0) = ω2 (0) = 0, where X = C02+α (Ω) × C02+α (Ω) with C02+α (Ω) = {u ∈ C 2+α (Ω) : ∂u ∂ν |∂Ω = 0} and ω1 , ω2 are in the orthogonal complement space of the variational matrix of the steady-state problem of (1.1) at U ∗ . As the proof of Theorem 6.1 in [12], if we let u(s) = a1−p + sφi0 + sω1 (s), v(s) = i0 T ai0 +sφi0 +sω2 (s), ai0 (s) = ai0 +β(s), U (s) = (u(s), v(s)) with φi0 being an eigenfunction of −∆ corresponding to µi0 subject to homogenous Neumann boundary condition, then (U (s); ai0 (s)) is the unique positive solution of the steady-state problem of model (1.1) near (Ui∗0 ; ai0 ). For the convenience of discussion, we denote by GU (U ∗ ; a), GU (U (s); ai0 (s)) the variational matrixes of the steady-state problem of (1.1) at (U ∗ ; a), (Ui∗0 ; ai0 ) respectively and (φ, ψ)T ∈ X the generator of the kernel space of GU (U (s); ai0 (s)).
3
The stability of U (s) = (u(s), v(s))T reads as follows.
∫ ∫ > 0 and Ω φdx, Ω φi0 ψdx ̸= 0. Then ∫ ∫ (U (s); ai0 (s)) is asymptotically stable if φdx and Ω Ω φi0 ψdx have the different signs; ∫ ∫ Whereas it is unstable if Ω φdx and Ω φi0 ψdx have the same sign. Theorem 2.2. Assume
3
pap−1 (1−d2 µi0 ) (d1 +d2 )µi0 +ap −p+1
Proof of Theorem 2.2
The proof of Theorem 6.1 [12] shows that 0 is an i-simple eigenvalue of GU (Ui∗0 ; ai0 ). According to Lemma 13.7 in [17], there exist smooth functions a → γ(a) and s → η(s) in the neighbourhoods of ai0 and 0 respectively such that γ(a), η(s) are the eigenvalues of GU (U ∗ ; a), GU (U (s); ai0 (s)) respectively, moreover, γ(a), η(s) satisfy γ(ai0 ) = 0 = η(0), γ ′ (ai0 ) ̸= 0. Thus, there is i ≥ 0 such that γ 2 − ((d1 + d2 )µi + ap − p + 1)γ + d1 d2 µ2i − (d1 (p − 1) − ap d2 )µi + ap = 0.
Differentiating on a and then taking a = ai0 , we have γ ′ (ai0 ) = using γ(ai0 ) = 0. Then Theorem 13.8 in [17] implies
pap−1 (1−d2 µi0 ) (d1 +d2 )µi0 +ap −p+1
> 0 by
sa′i0 (s)γ ′ (ai0 ) = −1. s→0 η(s) lim
Thus η(s) = −sa′i0 (s)γ ′ (ai0 )(1 + o(1)) = −sβ ′ (s)γ ′ (ai0 )(1 + o(1)), and the sign of η(s) is determined by β ′ (s). In the following, we focus on the sign of β ′ (s). By G(U (s); ai0 (s)) = 0 and ai0 (s) = ai0 + β(s), one obtains −d1 ∆u(s) + u(s)v p (s) − ai0 − β(s) = 0.
(3.1)
Multiply (3.1) by φ and then integrate over Ω, which leads to ∫ ∫ ∫ ∫ p − d1 ∆u(s)φdx + u(s)v(s) φdx − ai0 φdx − β(s) φdx ∫Ω ∫Ω ∫Ω ∫Ω u(s)v(s)p φdx − ai0 φdx − β(s) φdx = − d1 ∆φu(s)dx + Ω Ω Ω ∫ ∫ Ω ∫ ∫ p p−1 p u(s)v(s) φdx − ai0 φdx − β(s) φdx = − (v(s) φ + pu(s)v(s) ψ)u(s)dx + Ω
Ω
Ω
= 0.
Differentiating this equality on s we get ∫ ∫ ′ β (s) φdx = − (u′ (s)v(s)p + pu(s)v(s)p−1 v ′ (s))φdx Ω ∫Ω − (2pu(s)u′ (s)v(s)p−1 + p(p − 1)u(s)2 v(s)p−2 v ′ (s))ψdx ∫Ω + (u′ (s)v(s)p + pu(s)v(s)p−1 v ′ (s))φdx ∫Ω = − (2pu(s)u′ (s)v(s)p−1 + p(p − 1)u(s)2 v(s)p−2 v ′ (s))ψdx. Ω
4
Ω
′ ′ Since u(0) = a1−p i0 , v(0) = ai0 , u (0) = φi0 , v (0) = φi0 , we have ∫ ∫ −p ′ β (0) φdx = −(2p + p(p − 1)ai0 ) φi0 ψdx. Ω
If
∫
∫
Ω φdx, Ω φi0 ψdx
Ω
̸= 0, then ) (2p + p(p − 1)a−p ∫ i0 β (0) = − Ω φdx ′
∫
Ω φi0 ψdx
.
∫ ∫ Therefore, β ′ (0) > (<)0 if Ω φdx and Ω (2p + p(p − 1)a−p i0 )φi0 ψdx have the different signs (same sign), and further, β ′ (s) > (<)0, which means η(s) < (>)0 for small s > 0, and (U (s); ai0 (s)) is asymptotically stable (unstable). The proof is completed. ∫ Remark. In Theorem 2.2, we suppose that Ω φi0 ψdx ̸= 0, which is a key condition. ∫ However, it is not always so. That is, Ω φi0 ψdx = 0 may hold sometimes (for example, ∫1 in the one dimensional case, say, Ω = (0, 1), φi0 = cos(i0 πx), then 0 φi0 ψdx = 0 if ψ is a constant). If β ′ (0) = 0, then, for small s > 0, the sign of β ′ (s) is unknown, and further, the sign of η(s) is uncertain, which leads to the stability of (u(s), v(s))T is also unknown. In this case, it needs to change the idea to consider the stability of (u(s), v(s))T . Here we endow β(s) with stronger smoothness. In the following, we consider β ′′ (0) and present our idea on determining the stability of (u(s), v(s))T . First, we have ∫ ∫ β ′′ (s) φdx = −2p ((u′ (s))2 v(s)p−1 + u(s)u′′ (s)v(s)p−1 Ω
Ω
+(p − 1)u(s)u′ (s)v(s)p−2 v ′ (s))ψdx ∫ −p(p − 1) (u(s)v(s)p−2 u′ (s)v ′ (s) + (p − 2)(u(s))2 v(s)p−3 (v ′ (s))2 Ω
+(u(s)) v(s)p−2 v ′′ (s))ψdx. 2
Using u′′ (0) = 2ω1′ (0), v ′′ (0) = 2ω2′ (0), we have ∫ ∫ p ′′ 2 ′ β (0) φdx = −2p (a−1 i0 (ai0 + p − 1)φi0 + 2ω1 (0))ψdx Ω Ω ∫ −p 2 −p ′ −p(p − 1) (a−1 i0 (2 + (p − 2)ai0 )φi0 + 2ai0 ω2 (0))ψdx.
(3.2)
Ω
By using G(U (s); ai0 (s)) = 0 again, we obtain p −sd1 △(φi0 + ω1 (s)) + (a1−p i0 + sφi0 + sω1 (s))(ai0 + sφi0 + sω2 (s)) − ai0 (s) = 0, (3.3) p −sd2 △(φi0 + ω2 (s)) − (a1−p i0 + sφi0 + sω1 (s))(ai0 + sφi0 + sω2 (s)) + ai0 + sφi0 + sω2 (s) = 0. (3.4)
5
Differentiate (3.3) on s to get −d1 µi0 φi0 − d1 △ω1 (s) − sd1 △ω1′ (s) + (φi0 + ω1 (s) + sω1′ (s))(ai0 + sφi0 + sω2 (s))p
+p(ai1−p + sφi0 + sω1 (s))(ai0 + sφi0 + sω2 (s))p−1 (φi0 + ω2 (s) + sω2′ (s)) − a′i0 (s) = 0. 0
Taking s = 0, we have −d1 △ω1 (0) − (d1 µi0 − api0 − p)φi0 = 0.
(3.5)
Similarly, differentiate (3.4) on s and then also take s = 0 to yield −d2 △ω2 (0) − (d2 µi0 + api0 + p − 1)φi0 = 0.
(3.6)
Solving (3.5) and (3.6), ω1′ (0) and ω2′ (0) can be given. Then, substitute ω1′ (0) and ω2′ (0) into (3.2), and the sign of β ′′ (0) is determined. If β ′′ (0) > (<)0, then for small s > 0, β ′′ (s) > (<)0, which shows that β ′ (s) is increasing in s. While β ′ (0) = 0, then β ′ (s) > (<)0, which implies η(s) < (>)0, and (U (s); ai0 (s)) is asymptotically stable (unstable). Particularly, in the one dimensional case Ω = (0, 1), by (3.5), (3.6), it can be easily obtained ∫ p ω1′ (0) = (d−1 1 (ai0 − p) − µi0 )
1
0
and
ω2′ (0)
= −(µi0 +
p d−1 2 (ai0
+ p − 1))
φi0 dx = 0
∫
0
1
φi0 dx = 0,
respectively. Thus, we have ∫ ∫ ∫ −p −1 p −1 ′′ 2 β (0) φdx = −2pai0 (ai0 + p − 1) φi0 ψdx − p(p − 1)ai0 (2 + (p − 2)ai0 ) φ2i0 ψdx Ω Ω Ω ∫ p −p −1 2 −1 2 = −pai0 ((2ai0 − (3p − 2)ai0 + p ai0 + 4(p − 1))) φi0 ψdx. Ω
And further, the sign of β ′′ (0) is determined indeed.
References [1] G. Nicolis, Patterns of spatio-temporal organization in chemical and biochemical kinetics, SIAM-AMS Proc. 8 (1974), 33-58. [2] T.K. Callahan, E. Knobloch, Pattern formation in three-dimensional reactiondiffusion systems, Phys. D 132 (1999), 339-362. [3] P. Gray, S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C, Chem. Eng. Sci. 39 (1984), 1087-1097.
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[4] J.K. Hale, L.A. Peletier, W.C. Troy, Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis, SIAM J. Appl. Math. 61 (2000), 102-130. [5] Y. Li, Y. Wu, Stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, SIAM J. Math. Anal. 44 (2012), 14741521. [6] A.H. Msmali, M.I. Nelson, M.P. Edwards, Quadratic autocatalysis with non-linear decay, J. Math. Chem. 52 (2014), 2234-2258. [7] J. Billingham, D.J. Needham, A note on the properties of a family of travelling wave solutions arising in cubic autocatalysis, Dyn. Stab. Syst. 6 (1991), 33-49. [8] V. Gaspar, M. T. Beck, Depressing the bistable behavior of the iodate-arsenous acid reaction in a continuous flow stirred tank reactor by the effect of chloride or bromide ions: A method for determination of rate constants, J. Phys. Chem. 90 (1986), 63036305. [9] J.D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. [10] A.B. Finlayson, J.H. Merkin, Creation of spatial structure by an electric field applied to an ionic cubic autocatalator system, J. Engrg. Math. 38 (2000), 279-296. [11] G. Guo, B. Li, M. Wei, et al., Hopf bifurcation and steady-state bifurcation for an autocatalysis reaction-diffusion model, J. Math. Anal. Appl. 391 (2012), 265-277. [12] Y. Jia, Y. Li, J. Wu, Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics, Discrete Contin. Dyn. Syst. Ser. A 37 (2017), 4785-4813. [13] M.A.J. Chaplain, J.I. Tello, On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett. 57 (2016), 1-6. [14] S. Wu, Y. Song, Stability and spatiotemporal dynamics in a diffusive predator-prey model with nonlocal prey competition, Nonlinear Anal. Real World Applications 48 (2019), 12-39. [15] Y, Bai, Y. Li, Stability and Hopf bifurcation for a stage-structured predator-prey model incorporating refuge for prey and additional food for predator, Adv. Difference Equ. 2019 (2019), 42, 20pp. [16] Y. Jia, Computational analysis on Hopf bifurcation and stability for a consumerresource model with nonlinear functional response, Nonlinear Dyn. 94 (2018), 185195. [17] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-verlag, New York, 1999. [18] M.G. Crandall, P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal. 52 (1973), 161-180.
7
Jiantang Zhao, Yunfeng Jia
PII: DOI: Reference:
S0893-9659(19)30128-4 https://doi.org/10.1016/j.aml.2019.03.027 AML 5840
To appear in:
Applied Mathematics Letters
Received date : 30 November 2018 Revised date : 20 March 2019 Accepted date : 20 March 2019 Please cite this article as: J. Zhao and Y. Jia, Stability analysis on steady-state bifurcation for arbitrary order autocatalytic reaction model, Applied Mathematics Letters (2019), https://doi.org/10.1016/j.aml.2019.03.027 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Stability analysis on steady-state bifurcation for arbitrary order autocatalytic reaction model ∗ Jiantang Zhaoa , Yunfeng Jiab† a College of Mathematics and Information Science, Xianyang Normal University, Xianyang, Shaanxi 712000, China b School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710062, China
Abstract This paper deals with an autocatalytic reaction-diffusion model with arbitrary order functional response subject to no-flux boundary conditions. We mainly discuss the stability of the steady-state bifurcation which emanates from the unique positive constant equilibrium. On the stability of the bifurcation solution, the conventional way is to consider the sign of the first derivative of a certain function. However, sometimes, the first derivative may be equal to zero. This leads to the uncertainty of the stability. In such case, it needs to break through the common idea. We present an approach which determines the stability of the bifurcation solution. Keywords Autocatalytic reaction model; Steady-state bifurcation; Stability. Classifications 35J55, 35K57, 35Q80.
1
Introduction
In numerous chemical processes, when the reactions are far away from the thermodynamic equilibrium, the resemble behaviors between dynamical instability and the equilibrium phase changing on the steady-states in certain systems may likely occur. These behaviors are usually very important dynamical properties of chemical reactions and have been paid much attentions in recent decades. By analyzing these dynamics mathematically, people find that the reactions emerge tremendous fluctuation near the critical points, and then the chemical oscillation occurs as the reaction going on, which may cause the reactions to slow down, and then the system may give rise to critical transition to pattern formation when the phases are changed [1-2]. To study the chemical oscillation in chemical reaction-diffusion systems, it is important to master the kinetic generalities of these ∗
The work is supported in part by the Natural Science Foundations of China (11771262, 61672021), and by the Natural Science Basic Research Plan in Shaanxi Province of China (2018JQ1020). † Corresponding and co-first author.
1
systems near the critical points. These are the main concerns of autocatalytic reaction models, which have been applied to various problems in chemistry and biology, see [3-6] and the references therein for example. The reaction mechanism of the simplest autocatalytic reaction can be given by A + B → 2B, where A and B represent the reactant and autocatalyst, respectively. This reaction is one in which a molecule of species A interacts with a molecule of species B. An A molecule is converted into a B molecule. The final product consists of the original B molecule plusing the B molecule created in the reaction. A more complicated autocatalytic reaction is given by P → A,
A + 2B → 3B,
B → C,
where P is the density of the reaction precursor, A and B also represent the reactant and autocatalyst, C is some certain inert product. The supply of reactant is proceeded by the gradually degenerative process of the precursor. It is assumed that the reactant is effectively immobilised within the reactor, and the autocatalyst is made to flow through the reactor with a constant velocity as well as being able to diffuse, see [7] for details. Corresponding to this reaction mechanism, much of the previous work is concerned with − the iodate-arsenous acid systems, specifically, A and B represent IO− 3 and I , respectively [8]. The basic idea of this type reaction is that the diffusion of the autocatalyst has a stabilizing effect on the planar waves, whereas the diffusion of the substrate has a destabilizing effect. More complex autocatalytic reactions are involved in metabolic pathways and metabolic networks in biological systems. In this paper, we consider the simplified and non-dimensional form of arbitrary order autocatalytic model ut − d1 ∆u = a − uv p , x ∈ Ω, t > 0, p vt − d2 ∆v = uv − v, x ∈ Ω, t > 0, (1.1) ∂u ∂v = = 0, x ∈ ∂Ω, t > 0, ∂ν ∂ν u = u0 ≥, ̸≡ 0, v = v0 ≥, ̸≡ 0, x ∈ Ω, t = 0,
where Ω ⊂ Rn is bounded with smooth boundary ∂Ω, the variables u and v represent the dimensionless concentrations or densities of the reactant and autocatalyst respectively and so are usually assumed to be non-negative. d1 and d2 are the diffusion rates of u and v, respectively, a is a parameter representing the initial concentration of the reaction precursor, p ≥ 2 denotes the order of the reaction with respect to the autocatalytic species, ∂ a, d1 and d2 are positive constants. ∂ν denotes the differential in the direction of the outer normal to ∂Ω. The chemical reaction is driven within a closed container. For the derivation of model (1.1), we refer the readers to [9-10] and references therein for a more detailed description. In [11], Guo et al. considered the steady-state problem corresponding to model (1.1) in one dimensional case, specifically, Ω = (0, π). They mainly investigated the existence of Hopf bifurcation and the local steady-state bifurcation emitting from simple and double
2
eigenvalues. Then in [12], Jia et al. discussed the local steady-state bifurcation emitting from the unique positive constant equilibrium in high dimensional case and for p = 2. However, the stability of the steady-state bifurcation solution was not involved in both references. With what in mind, in this paper, we mainly deal with the stability of the local bifurcation solution bifurcating from the unique positive constant equilibrium U ∗ = (a1−p , a)T to the steady-state problem corresponding (1.1), since the stability of the steadystates is a very important property of dynamical systems which determines the progress of chemical reactions or the population evolution in a large extent (actually, there have been a large number of literatures focusing on the stability of different kinds of dynamical systems, such as [1-2, 5, 8, 12-16], etc). We use the conventional approach introduced in [17] when we consider the sign of the first derivative of a certain function. However, it should be pointed out that an idea is also presented when there is trouble in using the first derivative, which is analyzed and clarified after the proof of the main theorem.
2
Main results
Let µi , i = 0, 1, 2, · · · be all eigenvalues of the operator −∆ with homogenous Neumann boundary conditions. By using the very similar techniques as in proving Theorem 6.1 in [12], and combining with Theorem 3.1 (i) in [11], the following result can be easily obtained, and so we omit the proof here. Theorem 2.1. Assume that there is i > 0 satisfying d2 µi < p − 1. Set ai =
(
d1 µi (p − 1 − d2 µi ) 1 + d2 µi
)1
p
,
i = 1, 2, · · · .
Suppose that the followings hold. (i) ai ̸= aj for any integer i ̸= j; (ii) a = ai0 for some i0 ; (iii) µi0 is simple. Then (Ui∗0 ; ai0 ) is a bifurcation point to the steady-state problem of (1.1), where Ui∗0 = T (a1−p i0 , ai0 ) is the unique positive constant equilibrium to (1.1). Following Theorem 2.1 and according to the Crandall-Rabinowitz bifurcation theorem [18], we know that there exist δ > 0 and smooth functions β : (−δ, δ) −→ R, (ω1 , ω2 )T : (−δ, δ) −→ X satisfying β(0) = 0, ω1 (0) = ω2 (0) = 0, where X = C02+α (Ω) × C02+α (Ω) with C02+α (Ω) = {u ∈ C 2+α (Ω) : ∂u ∂ν |∂Ω = 0} and ω1 , ω2 are in the orthogonal complement space of the variational matrix of the steady-state problem of (1.1) at U ∗ . As the proof of Theorem 6.1 in [12], if we let u(s) = a1−p + sφi0 + sω1 (s), v(s) = i0 T ai0 +sφi0 +sω2 (s), ai0 (s) = ai0 +β(s), U (s) = (u(s), v(s)) with φi0 being an eigenfunction of −∆ corresponding to µi0 subject to homogenous Neumann boundary condition, then (U (s); ai0 (s)) is the unique positive solution of the steady-state problem of model (1.1) near (Ui∗0 ; ai0 ). For the convenience of discussion, we denote by GU (U ∗ ; a), GU (U (s); ai0 (s)) the variational matrixes of the steady-state problem of (1.1) at (U ∗ ; a), (Ui∗0 ; ai0 ) respectively and (φ, ψ)T ∈ X the generator of the kernel space of GU (U (s); ai0 (s)).
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The stability of U (s) = (u(s), v(s))T reads as follows.
∫ ∫ > 0 and Ω φdx, Ω φi0 ψdx ̸= 0. Then ∫ ∫ (U (s); ai0 (s)) is asymptotically stable if φdx and Ω Ω φi0 ψdx have the different signs; ∫ ∫ Whereas it is unstable if Ω φdx and Ω φi0 ψdx have the same sign. Theorem 2.2. Assume
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pap−1 (1−d2 µi0 ) (d1 +d2 )µi0 +ap −p+1
Proof of Theorem 2.2
The proof of Theorem 6.1 [12] shows that 0 is an i-simple eigenvalue of GU (Ui∗0 ; ai0 ). According to Lemma 13.7 in [17], there exist smooth functions a → γ(a) and s → η(s) in the neighbourhoods of ai0 and 0 respectively such that γ(a), η(s) are the eigenvalues of GU (U ∗ ; a), GU (U (s); ai0 (s)) respectively, moreover, γ(a), η(s) satisfy γ(ai0 ) = 0 = η(0), γ ′ (ai0 ) ̸= 0. Thus, there is i ≥ 0 such that γ 2 − ((d1 + d2 )µi + ap − p + 1)γ + d1 d2 µ2i − (d1 (p − 1) − ap d2 )µi + ap = 0.
Differentiating on a and then taking a = ai0 , we have γ ′ (ai0 ) = using γ(ai0 ) = 0. Then Theorem 13.8 in [17] implies
pap−1 (1−d2 µi0 ) (d1 +d2 )µi0 +ap −p+1
> 0 by
sa′i0 (s)γ ′ (ai0 ) = −1. s→0 η(s) lim
Thus η(s) = −sa′i0 (s)γ ′ (ai0 )(1 + o(1)) = −sβ ′ (s)γ ′ (ai0 )(1 + o(1)), and the sign of η(s) is determined by β ′ (s). In the following, we focus on the sign of β ′ (s). By G(U (s); ai0 (s)) = 0 and ai0 (s) = ai0 + β(s), one obtains −d1 ∆u(s) + u(s)v p (s) − ai0 − β(s) = 0.
(3.1)
Multiply (3.1) by φ and then integrate over Ω, which leads to ∫ ∫ ∫ ∫ p − d1 ∆u(s)φdx + u(s)v(s) φdx − ai0 φdx − β(s) φdx ∫Ω ∫Ω ∫Ω ∫Ω u(s)v(s)p φdx − ai0 φdx − β(s) φdx = − d1 ∆φu(s)dx + Ω Ω Ω ∫ ∫ Ω ∫ ∫ p p−1 p u(s)v(s) φdx − ai0 φdx − β(s) φdx = − (v(s) φ + pu(s)v(s) ψ)u(s)dx + Ω
Ω
Ω
= 0.
Differentiating this equality on s we get ∫ ∫ ′ β (s) φdx = − (u′ (s)v(s)p + pu(s)v(s)p−1 v ′ (s))φdx Ω ∫Ω − (2pu(s)u′ (s)v(s)p−1 + p(p − 1)u(s)2 v(s)p−2 v ′ (s))ψdx ∫Ω + (u′ (s)v(s)p + pu(s)v(s)p−1 v ′ (s))φdx ∫Ω = − (2pu(s)u′ (s)v(s)p−1 + p(p − 1)u(s)2 v(s)p−2 v ′ (s))ψdx. Ω
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Ω
′ ′ Since u(0) = a1−p i0 , v(0) = ai0 , u (0) = φi0 , v (0) = φi0 , we have ∫ ∫ −p ′ β (0) φdx = −(2p + p(p − 1)ai0 ) φi0 ψdx. Ω
If
∫
∫
Ω φdx, Ω φi0 ψdx
Ω
̸= 0, then ) (2p + p(p − 1)a−p ∫ i0 β (0) = − Ω φdx ′
∫
Ω φi0 ψdx
.
∫ ∫ Therefore, β ′ (0) > (<)0 if Ω φdx and Ω (2p + p(p − 1)a−p i0 )φi0 ψdx have the different signs (same sign), and further, β ′ (s) > (<)0, which means η(s) < (>)0 for small s > 0, and (U (s); ai0 (s)) is asymptotically stable (unstable). The proof is completed. ∫ Remark. In Theorem 2.2, we suppose that Ω φi0 ψdx ̸= 0, which is a key condition. ∫ However, it is not always so. That is, Ω φi0 ψdx = 0 may hold sometimes (for example, ∫1 in the one dimensional case, say, Ω = (0, 1), φi0 = cos(i0 πx), then 0 φi0 ψdx = 0 if ψ is a constant). If β ′ (0) = 0, then, for small s > 0, the sign of β ′ (s) is unknown, and further, the sign of η(s) is uncertain, which leads to the stability of (u(s), v(s))T is also unknown. In this case, it needs to change the idea to consider the stability of (u(s), v(s))T . Here we endow β(s) with stronger smoothness. In the following, we consider β ′′ (0) and present our idea on determining the stability of (u(s), v(s))T . First, we have ∫ ∫ β ′′ (s) φdx = −2p ((u′ (s))2 v(s)p−1 + u(s)u′′ (s)v(s)p−1 Ω
Ω
+(p − 1)u(s)u′ (s)v(s)p−2 v ′ (s))ψdx ∫ −p(p − 1) (u(s)v(s)p−2 u′ (s)v ′ (s) + (p − 2)(u(s))2 v(s)p−3 (v ′ (s))2 Ω
+(u(s)) v(s)p−2 v ′′ (s))ψdx. 2
Using u′′ (0) = 2ω1′ (0), v ′′ (0) = 2ω2′ (0), we have ∫ ∫ p ′′ 2 ′ β (0) φdx = −2p (a−1 i0 (ai0 + p − 1)φi0 + 2ω1 (0))ψdx Ω Ω ∫ −p 2 −p ′ −p(p − 1) (a−1 i0 (2 + (p − 2)ai0 )φi0 + 2ai0 ω2 (0))ψdx.
(3.2)
Ω
By using G(U (s); ai0 (s)) = 0 again, we obtain p −sd1 △(φi0 + ω1 (s)) + (a1−p i0 + sφi0 + sω1 (s))(ai0 + sφi0 + sω2 (s)) − ai0 (s) = 0, (3.3) p −sd2 △(φi0 + ω2 (s)) − (a1−p i0 + sφi0 + sω1 (s))(ai0 + sφi0 + sω2 (s)) + ai0 + sφi0 + sω2 (s) = 0. (3.4)
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Differentiate (3.3) on s to get −d1 µi0 φi0 − d1 △ω1 (s) − sd1 △ω1′ (s) + (φi0 + ω1 (s) + sω1′ (s))(ai0 + sφi0 + sω2 (s))p
+p(ai1−p + sφi0 + sω1 (s))(ai0 + sφi0 + sω2 (s))p−1 (φi0 + ω2 (s) + sω2′ (s)) − a′i0 (s) = 0. 0
Taking s = 0, we have −d1 △ω1 (0) − (d1 µi0 − api0 − p)φi0 = 0.
(3.5)
Similarly, differentiate (3.4) on s and then also take s = 0 to yield −d2 △ω2 (0) − (d2 µi0 + api0 + p − 1)φi0 = 0.
(3.6)
Solving (3.5) and (3.6), ω1′ (0) and ω2′ (0) can be given. Then, substitute ω1′ (0) and ω2′ (0) into (3.2), and the sign of β ′′ (0) is determined. If β ′′ (0) > (<)0, then for small s > 0, β ′′ (s) > (<)0, which shows that β ′ (s) is increasing in s. While β ′ (0) = 0, then β ′ (s) > (<)0, which implies η(s) < (>)0, and (U (s); ai0 (s)) is asymptotically stable (unstable). Particularly, in the one dimensional case Ω = (0, 1), by (3.5), (3.6), it can be easily obtained ∫ p ω1′ (0) = (d−1 1 (ai0 − p) − µi0 )
1
0
and
ω2′ (0)
= −(µi0 +
p d−1 2 (ai0
+ p − 1))
φi0 dx = 0
∫
0
1
φi0 dx = 0,
respectively. Thus, we have ∫ ∫ ∫ −p −1 p −1 ′′ 2 β (0) φdx = −2pai0 (ai0 + p − 1) φi0 ψdx − p(p − 1)ai0 (2 + (p − 2)ai0 ) φ2i0 ψdx Ω Ω Ω ∫ p −p −1 2 −1 2 = −pai0 ((2ai0 − (3p − 2)ai0 + p ai0 + 4(p − 1))) φi0 ψdx. Ω
And further, the sign of β ′′ (0) is determined indeed.
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