- Email: [email protected]
On steady-state solutions of the Brusselator-type system
Nonlinear Analysis 71 (2009) 1389–1394
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
On steady-state solutions of the Brusselator-type systemI Rui Peng a,b,∗ , Ming Yang c a
Institute of Nonlinear Complex Systems, College of Science, China Three Gorges University, Yichang City, 443002, Hubei Province, PR China
b
School of Science and Technology, University of New England, Armidale, NSW, 2351, Australia
c
Department of Mathematics, Southeast University, Nanjing, 210018, PR China
article
info
Article history: Received 14 October 2008 Accepted 1 December 2008 MSC: 35J55 35B45 35K57 92C40 Keywords: Brusselator-type system Steady state Existence Non-existence Asymptotic behavior
a b s t r a c t In this article, we shall be concerned with the following Brusselator-type system:
−θ ∆u = λ(1 − (b + 1)u + bum v) −∆v = λa2 (u − um v)
in Ω , in Ω ,
under the homogeneous Neumann boundary conditions. This system was recently investigated by M. Ghergu in [Nonlinearity, 21 (2008), 2331–2345]. Here, Ω ⊂ RN (N ≥ 1) is a smooth and bounded domain and a, b, m, λ and θ are positive constants. When m = 2, this system corresponds to the well-known stationary Brusselator model which has received extensive studies analytically as well as numerically. In the present work, we derive some further results for the general system. Our conclusions show that there is no non-constant positive steady state for large a while small a may produce non-constant positive steady states. If 1 ≤ N ≤ 3 and 1 < m < 3, we particularly determine the asymptotic behavior of non-constant positive steady states as a converges to zero, thereby solving an open problem left in Ghergu’ work. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction In this article, after a scaling as in [1,2], we shall be concerned with the following general stationary Brusselator system:
−θ ∆u = λ(1 − (b + 1)u + bum v) −∆v = λa2 (u − um v) ∂ u = ∂v = 0 ∂ν ∂ν
in Ω , in Ω ,
(1.1)
on ∂ Ω .
Here, Ω ⊂ RN is a bounded domain with smooth boundary ∂ Ω , ν is the outward unit normal vector on ∂ Ω . The unknown functions u(x) and v(x) represent the spatial concentrations of two intermediate reactants and are considered to be nonnegative, a and b are fixed concentrations of other components, θ is the diffusion coefficient of the concentration u, and λ is a measure of the size of the domain. Therefore, a, b, θ and λ are always assumed to be positive constants. For more interpretation of the background of this chemical model, the interested reader may further refer to [3–5,1,2,6].
I This work was partially supported by the National Natural Science Foundation of China (10801090, 10871185, 10726016, 10601011), the Scientific Research Projects of Hubei Provincial Department of Education (Q200713001) and Scientific Innovation Team Project of Hubei Provincial Department of Education (T200809). ∗ Corresponding address: Department of Mathematics, College of Science, China Three Gorges University, Yichang City, 443002, Hubei Province, PR China. E-mail addresses: [email protected] (R. Peng), [email protected] (M. Yang).
0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.12.003
1390
R. Peng, M. Yang / Nonlinear Analysis 71 (2009) 1389–1394
When m = 2, (1.1) becomes the well-known Brusselator model, which was proposed by Prigogine and Lefever [7] in 1968 to model morphogenesis and pattern formation in chemical reactions. In recent decades, the Brusselator system has received both intensive analytical and numerical studies by many authors, for example, [3–5,2] and the references therein. The fascinating Turing instability and Turing pattern have been observed through this looking-simple coupled elliptic system. Moreover, the associated dynamics of the Brusselator model was dealt with by You in [6]. For the extended form of Brusselator model (1.1), Ghergu [1] also focused on the steady-state problem. His results exhibit the critical role the nonlinear term um plays in generating the non-constant positive solutions (i.e., stationary pattern) of (1.1). To be precise, he found that, if m ≤ 1, (1.1) admits no non-constant positive solution; on the contrary, if m > 1, among other proper conditions on a, b, θ , λ and Ω , (1.1) possesses at least one non-constant positive solution. He also obtained various existence and non-existence conclusions with respect to the parameters b, θ , λ. To achieve these results, in [1], Ghergu first applied the maximum principle (due to [8]) and integration estimates to establish upper and lower bounds for any classical positive solution to (1.1). Then, with the helps of various mathematical tools such as the energy integration, implicit function theorem and topological degree, the work of Ghergu provides lots of interesting criteria for the existence or non-existence for non-constant positive solutions of (1.1), some of which turn out to be the improvement of the results derived by Peng and Wang in [2]. It is clear that only nonnegative solutions of (1.1) are of physical interests. As in the existing works [3–5,1,2], we are also interested in the existence and non-existence of non-constant positive solution to (1.1). From now on, whenever mentioning the positive solution (u, v) of (1.1), we always mean that (u, v) ∈ C 2 (Ω ) × C 2 (Ω ) such that u, v are positive on Ω and satisfy (1.1). Additionally, it should be noted that (u, v) = (1, 1) is the unique constant positive solution of (1.1). For sake of the reader’s reference, we are ready to list the main results of [1] as follows. First of all, by the maximum principle for elliptic equations which comes from Lou and Ni (see, [8], Proposition 2.2 or [9] Lemma 2.1), without any restriction on spatial dimensions, Ghergu established the upper and lower bounds for arbitrary positive solution of (1.1). Theorem 1.1 (See Ghergu [1], Theorem 2.2). Let m > 1, then any positive solution (u, v) of (1.1) satisfies 1 b+1
≤u≤1+
b(b + 1)m−1
θ a2
and
1+
b(b + 1)m−1
θ a2
1−m
≤ v ≤ (b + 1)m−1 .
Then, based on Theorem 1.1 and some other a priori estimates, using several analytical techniques including the maximum principle, energy integration and the method of implicit function theorem (due to Peng and Wang [2] and Peng [10]), Ghergu deduced the following results concerning the non-existence of non-constant positive solutions to (1.1). Theorem 1.2 (See Ghergu [1], Section 3). The system (1.1) has no non-constant positive solution if one of the following cases happens: (i) 0 < m ≤ 1; (ii) λ is small enough; (iii) θ is large enough; (iv) b is small enough. Let 0 = µ0 < µ1 < µ2 < · · · be the eigenvalues of −∆ in Ω with the homogeneous Neumann boundary conditions, and let ei be the algebraic multiplicity of µi . Thus, combining Theorems 1.1 and 1.2, and the calculation formula of Leray–Schauder degree and its invariance with respect to homotopy (see [11]), the author used the argument similar to Theorem 5.1 of [2] to establish some existence results regarding non-constant positive solutions to (1.1). In particular, he showed Theorem 1.3 (See Ghergu [1], Corollary 4.3). Let λ, θ > 0 be fixed and b(m − 1) > 1. Assume that for some i ≥ 1,
λθ −1 (b(m − 1) − 1) ∈ (µi , µi+1 ) and
i X
ek is odd,
(1.2)
k=1
then there exists a A > 0 such that (1.1) has at least one non-constant positive solution for all 0 < a < A. The main purpose of the present work is to derive some further results for (1.1). Firstly, we shall prove the non-existence for non-constant positive solutions to (1.1) in two cases: (1) m is larger than but close to 1; (2) a is sufficiently large. To this end, we apply the implicit function theorem and topological degree argument. More precisely, our first result can be summarized as follows. Theorem 1.4. (i) Let a, b, θ and λ be fixed, then there exists a small 0 > 0 such that (1.1) has no non-constant positive solution if m < 1 + 0 ; (ii) Let b, m, θ and λ be fixed, then there exists a large a > 0 such that (1.1) has no non-constant positive solution if a > a. As the second result, we attempt to determine the asymptotic profile of the non-constant positive solutions obtained in Theorem 1.3 as a converges to zero. For emphasizing the dependence of these non-constant positive solutions (u, v) on a, we denote them to be (ua , va ). It was conjectured in the concluding section of [1] that, as a goes to zero, va approaches a constant profile while the limit of ua satisfies a nonlinear elliptic equation with the homogeneous Neumann boundary condition. Our
R. Peng, M. Yang / Nonlinear Analysis 71 (2009) 1389–1394
1391
statement will give an affirmative answer to this open problem. The key point of our analysis is to establish more delicate a priori estimates for positive solutions to (1.1) as a → 0. However, due to mathematical difficulties, in proving the desired assertion we have to restrict 1 ≤ N ≤ 3; and if N = 3, m also satisfies 1 < m < 3. Taking into consideration the realistic situation of chemistry, the restriction 1 ≤ N ≤ 3 is enough for the possible application of the extended Brusselator system (especially for the Brusselator system with m = 2). Theorem 1.5. Let λ, θ > 0 be fixed, and b(m − 1) > 1 and 1 ≤ N ≤ 3. If N = 3, the constant m also satisfies 1 < m < 3. Assume that for some i ≥ 1, the condition (1.2) is also satisfied. Then, for any sequence of non-constant positive solution (ua , va ) to (1.1), there is a subsequence (uan , van ) of (ua , va ) satisfying that (uan , van ) converges to (u∗ , v ∗ ) on C 2 (Ω )× C 2 (Ω ) as an → 0+ , where u∗ is a non-constant positive solution of
" − θ ∆u = λ 1 − (b + 1)u + bu
m
Z
m
#
−1 Z
u dx
udx
Ω
Ω
in Ω ,
∂u =0 ∂ν
on ∂ Ω ,
(1.3)
and v ∗ is uniquely determined by
v∗ =
Z Ω
(u∗ )m dx
−1 Z Ω
u∗ dx.
Therefore, if (1.2) holds, the non-local elliptic problem (1.3) admits at least one non-constant positive solution. We would like to point out that, in general, it is very hard to prove the existence of non-constant positive solutions of the non-local elliptic equation such as (1.3). In terms of chemistry, Theorem 1.5 implies that, when the concentration of a certain component (represented by a) is small enough, then in the two intermediate reactants, the component v must be spatially homogeneous while the other component u may be spatially heterogeneous. 2. Proof of main results In this section, we shall give the detailed proof of Theorems 1.4 and 1.5. From now on, all the integrals are computed over
Ω . For any 1 ≤ p, we use k · kp and k · k2,p to stand for the usual norms of Banach spaces Lp (Ω ) and W 2,p (Ω ), respectively. As some preliminaries for the proof of Theorem 1.5, we need to recall two basic results. Assume that (u, v) is a nonnegative solution to (1.1), then integrating the equations in (1.1), we find
(b + 1)
Z
Z u−b
um v = |Ω | and
Z
Z u=
um v.
(2.1)
Here and in what follows, |Ω | always represents the volume of the domain Ω . Hence, it follows from (2.1) that
Z
u = |Ω |.
(2.2)
We will have to present a local result for weak super-solution of linear elliptic equations from [12] (also see, for example, [13], Theorem 8.18). Lemma 2.1. Let Ω be a bounded Lipschitz domain in RN . Let Λ be a non-negative constant and suppose that z ∈ W 1,2 (Ω ) is a non-negative weak solution of the inequalities 0 ≤ −∆z + Λz
in Ω ,
∂z = 0 on ∂ Ω . ∂ν
Then, for any q ∈ [1, N /(N − 2)), there exists a positive constant C0 , depending only on q, Λ and Ω , such that
kz kq ≤ C0 inf z . Ω
2.1. Proof of Theorem 1.4 Proof of (ii) of Theorem 1.4. Assume that (ua , va ) is an arbitrary positive solution of (1.1). Then, we first claim that (ua , va ) → (1, 1) uniformly on Ω as a → ∞. By Theorem 1.1, we know both ua and va are uniformly bounded on Ω for all large a. Then, from the standard regularity theory for elliptic equations and imbedding theorems, it follows from the first equation of (1.1) that there is a sequence an with an → ∞ as n → ∞ and the corresponding solution (un , vn ) =: (uan , van ), such that un → u˜ in C 1 (Ω ) as n → ∞. Furthermore, u˜ is a positive function on Ω . Since vn satisfies
−∆vn = λa2n (un − um n vn ) in Ω ,
∂vn = 0 on ∂ Ω , ∂ν
1392
R. Peng, M. Yang / Nonlinear Analysis 71 (2009) 1389–1394
we can proceed as in the proof of Lemma 3.4 of [14] to assert vn → (˜u)1−m uniformly on Ω as a → ∞. As a result, it is easy to see that u˜ is a solution of the following equation
−θ ∆u˜ = λ(1 − u˜ ) in Ω ,
∂ u˜ = 0 on ∂ Ω . ∂ν
This equation has a unique solution u˜ = 1. Therefore, our previous claim holds true. In order to achieve the desired conclusion, we need to use a different argument from those in [1,2] which seem not to work in our case. The argument here is essentially similar to that of the proof for Theorem 3.3 in [14]. We shall mainly apply the theory of topological degree. For this purpose, we shall have to reformulate the system (1.1) in the framework that the Leray–Schauder degree theory can work. Let us denote
Ξ = {(u, v) ∈ C (Ω ) × C (Ω ) : C < u, v < C }, where the positive numbers C and C satisfying 0 < C < 1 and 1 < C will be determined later. We also define the operator
Γ (a, b, u, v) = (−∆ + I)−1 u + θ −1 λ(1 − (b + 1)u + bum v), v + λa2 (u − um v) , where (−∆ + I)−1 stands for the inverse operator of −∆ + I subject to Neumann boundary condition over ∂ Ω . Let a < a and b0 be arbitrary positive constants. It is well-known that Γ is a compact operator from [a, a] × [0, b0 ] × Ξ to C (Ω ) × C (Ω ). Furthermore, (u, v) ∈ Ξ solves (1.1) if and only if (u, v) satisfies (u, v) = Γ (a, b, u, v). In addition, by the fact that (ua , va ) → (1, 1) uniformly on Ω as a → ∞, together with Theorem 1.1, we can find the positive numbers a, C and C , such that
(u, v) 6= Γ (a, b, u, v),
∀a ∈ [a, ∞), b ∈ [0, b0 ] and (u, v) ∈ ∂ Ξ .
As a result, the topological degree deg (I − Γ (a, b, ·), Ξ , 0) is well-defined and is also independent of a ∈ [a, ∞) and b ∈ [0, b0 ]. We recall that Theorem 1.1 with b = 0 implies that (1, 1) is the unique fixed point of Γ (a, 0, ·) in Ξ , and thus deg(I − Γ (a, 0, ·), Ξ , 0) = index (I − Γ (a, 0, ·), (1, 1)). On the other hand, the remark of Section 5 in [1] states that (1, 1) is linearly stable as the unique constant positive steady state for the corresponding reaction–diffusion system of (1.1) with (a, b) = (a, 0) if we take a to be larger if necessary. Hence, by the well-known Leray–Schauder degree formula (see, e.g., Theorem 2.8.1 in [11]), we have deg(I − Γ (a, 0, ·), Ξ , 0) = index (I − Γ (a, 0, ·), (1, 1)) = 1. Therefore, it follows that deg(I − Γ (a, b, ·), Ξ , 0) = deg (I − Γ (a, 0, ·), Ξ , 0) = 1, for any a ∈ [a, ∞) and b ∈ [0, b0 ]. We have shown that for any positive solution (ua , va ) of (1.1), (ua , va ) → (1, 1) uniformly on Ω as a → ∞. Therefore, according to the discussion of Section 5 in [1], carrying out the same elementary computation as in Theorem 2.1 of [2], we can take a larger a if necessary such that every possible positive solution (ua , va ) of (1.1) is linearly stable provided that a ≥ a. Hence, the fixed point index of Γ (a, b, ua , va ) is well-defined and is equal to 1 if a ≥ a and b0 ≥ b ≥ 0. Furthermore, for such fixed a and b, the compactness of Γ (a, b, ·) implies that there are at most finitely many such fixed points in Ξ , denoted by {(ui , vi )}`1 . Then, from the additivity property of the Leray–Schauder degree, it follows that 1 = deg(I − Γ (a, b, ·), Ξ , 0) =
` X
index (I − Γ (a, b, ·), (ui , vi )) = `.
1
This indicates the uniqueness of positive solutions of (1.1) for a ≥ a and so the unique positive solution must be (1, 1). Our proof is complete. The proof of (i) in Theorem 1.4 can be obtained similarly to the above but is simpler. It is sufficient to note that, for m = 1, (1.1) has a unique positive solution (1, 1) which is also linearly stable when m is close to 1. We leave the details to the interested reader. 2.2. Proof of Theorem 1.5 Proof. For sake of clarity, we divide our proof into two steps. Step 1. We first have to establish some a priori upper and lower estimates for positive solutions (ua , va ) of (1.1) as a → 0+ . In virtue of Theorem 1.1, it remains to derive the upper bound of ua and lower bound of va .
R. Peng, M. Yang / Nonlinear Analysis 71 (2009) 1389–1394
1393
We first show that kua k∞ is bounded from above as a → 0+ . Note that
∂ ua = 0 on ∂ Ω . ∂ν
−θ ∆ua + λ(b + 1)ua > 0 in Ω , Thus, by use of Lemma 2.1, we have
kua kq ≤ C0 inf ua .
(2.3)
Ω
Here, q ≥ 1 can take arbitrary large number if N = 1 or 2; and q ∈ [1, 3) if N = 3. Here and in what follows, the positive constant C0 depends only on b, m, λ, θ and Ω , and may vary from place to place. From (2.3), it is not hard to see that
k ua k q ≤ C 0 .
(2.4)
Otherwise, we can find a sequence {an } with an → 0 as n → ∞ and a corresponding positive solution sequence {(uan , van )}, such that kuan kq → ∞. Thus, from (2.3), it follows uan → ∞ uniformly on Ω as n → ∞. This is a contradiction with (2.2). We also observe that va ≤ (b + 1)m−1 on Ω . When N = 1 or 2, by taking q = 2m in (2.4), we find +
kλ(1 − (b + 1)ua + bum a va )k2 ≤ C0 . Then, in accordance with the equation of ua , the well-known Lp theory for elliptic equations gives kua k2,2 ≤ C0 . Hence, for N = 1, 2, the imbedding theorem W 2,2 (Ω ) ,→ C α (Ω ) for some α ∈ (0, 1) concludes kua k∞ ≤ C0 , as required. If N = 3, together with (2.4) for 1 ≤ q < 3, we obtain the following estimates:
kλ(1 − (b + 1)ua + bum a va )kp ≤ C0 , for 1 ≤ p <
3 m
.
Thanks to the Lp theory for elliptic equations again, one gets kua k2,p ≤ C0 . When 3 − 2p ≤ 0, the imbedding theorem W 2,p (Ω ) ,→ Lr (Ω ) for all 1 ≤ r < ∞ deduces kua kr0 ≤ C0 for some r0 > 3/2. Thus, using the equation of ua and the imbedding theorem W 2,r0 (Ω ) ,→ C α (Ω ) for some α ∈ (0, 1) again, we can easily obtain kua k∞ ≤ C0 . When 3 − 2p > 0, it follows from W 2,p (Ω ) ,→ Lr1 (Ω ) for any 1 ≤ r1 < 3p/(3 − 2p) that kua kr1 ≤ C0 for 1 ≤ r1 < 3p/(3 − 2p). Once again, we obtain kua k2,r1 ≤ C0 and so kua kr2 ≤ C0 for 1 ≤ r2 < 3r1 /(3 − 2r1 ). The condition 1 < m < 3 allows us to repeat the bootstrapping procedure so that we find a rn such that 3 − 2rn ≤ 0. In this case, we can continue to use the analysis of the above paragraph to assert kua k∞ ≤ C0 . Next, we verify the fact that there exists a positive lower bound for va for all small a. If this is false, we can find a sequence {an } with an → 0+ as n → ∞ and a corresponding positive solution sequence {(un , vn )}, such that infΩ vn → 0 as n → ∞. Combining the standard regularity theory for elliptic equations and imbedding theorems and the equations of un and vn , passing to a further sequence of {an } if necessary, we may assume that un → u∗ and vn → v ∗ in C 2 (Ω ), and u∗ > 0 on Ω due to Theorem 1.1. In addition, v ∗ ≥ 0 on Ω and satisfies
∂v ∗ = 0 on ∂ Ω , ∂ν
−∆v ∗ = 0 in Ω ,
which implies v ∗ ≡ 0 on Ω . On the other hand, by (2.1), we notice
Z lim
n→∞
Z un =
u∗ =
Z
(u∗ )m v ∗ = 0,
and so this arrives at a contradiction with (2.2). Until now, we have verified that there exist two positive numbers C1 and C2 which is independent of all small a, such that for any positive solution (ua , va ) of (1.1), the following holds: C1 ≤ ua (x), va (x) ≤ C2 ,
∀x ∈ Ω .
(2.5)
Step 2. Based on the estimates (2.5), we prove the asymptotic behavior of non-constant positive solutions obtained in Theorem 1.3. Suppose that our conclusion stated in Theorem 1.5 is not true. Then, using (2.5), the standard Lp and Schauder’s estimates and the imbedding theorems, there exists a sequence {an } with an → 0+ as n → ∞ and the corresponding nonconstant positive solution (un , vn ) to (1.1) for a = an , such that there is a subsequence of (un , vn ), still denoted by itself, ¯ ) × C 2 (Ω ¯ ) as n → ∞. satisfying (un , vn ) → (1, 1) on C 2 (Ω Since (un , vn ) is non-constant in Ω for each n ≥ 1, we can define hn = un − 1,
kn = vn − 1,
and h˜ n =
hn
khn k∞ + k k n k∞
,
k˜ n =
kn
khn k∞ + kkn k∞
.
1394
R. Peng, M. Yang / Nonlinear Analysis 71 (2009) 1389–1394
As a result, h˜ n and k˜ n satisfy
˜ −θ ∆h˜ n = −λ(b + 1)h˜ n + bmλζnm−1 h˜ n + bλum n kn 2˜ 2 m−1 ˜ 2 m˜ ˜ −∆kn = λan hn − mλan ζn hn − λan un kn ˜ ˜ ∂ hn = ∂ kn = 0 ∂ν ∂ν
in Ω , in Ω ,
(2.6)
on ∂ Ω ,
where ζn (x) lies between un (x) and 1. Applying the standard theory, up to a further sequence, we can assume that (h˜ n , k˜ n ) → ¯ ) × C 1 (Ω ¯ ) as n → ∞. By passing to the limit in (2.6), we find that (h˜ , k˜ ) in C 1 (Ω
−θ ∆h˜ = λ(bm − (b + 1))h˜ + bλk˜ ,
−∆k˜ = 0 in Ω ,
∂ h˜ ∂ k˜ = = 0 on ∂ Ω . ∂ν ∂ν
Hence, k˜ is a constant and so our hypothesis λθ −1 (bm − (b + 1)) 6= µn for any n ≥ 0 deduces
(bm − (b + 1))h˜ + bk˜ = 0.
(2.7)
Furthermore, kh˜ n k∞ + kk˜ n k∞ = 1 for each n ≥ 1 gives
|h˜ | + |k˜ | = 1. On the other hand, integrating the second equation in (2.6), dividing the resulting integral by
(2.8) a2n
and then letting n → ∞,
˜ which, together with (2.7) and (2.8), leads to a contradiction. Consequently, under the assumption we obtain (m − 1)h˜ = k, of Theorem 1.5, simple analysis asserts, for any sequence of non-constant positive solution (ua , va ) to (1.1), there is a subsequence (uan , van ) of (ua , va ) satisfying that (uan , van ) converges to (u∗ , v ∗ ) on C 2 (Ω ) × C 2 (Ω ) as an → 0+ , where u∗ and v ∗ are determined as in Theorem 1.5. Hence, the proof is now finished. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
M. Ghergu, Non-constant steady-state solutions for Brusselator type systems, Nonlinearity 21 (2008) 2331–2345. R. Peng, M.X. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl. 309 (2005) 151–166. K.J. Brown, F.A. Davidson, Global bifurcation in the Brusselator system, Nonlinear Anal. TMA 12 (1995) 1713–1725. T.K. Callahan, E. Knobloch, Pattern formation in three-dimensional reaction–diffusion systems, Physica D 132 (1999) 339–362. T. Erneux, E. Reiss, Brusselator isolas, SIAM J. Appl. Math. 43 (1983) 1240–1246. Y. You, Global dynamics of the Brusselator equations, Dyn. Partial Diff. Eqns. 4 (2007) 167–196. I. Prigogene, R. Lefever, Symmetry breaking instabilities in dissipative systems II, J. Chem. Phys. 48 (1968) 1665–1700. Y. Lou, W.M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations 131 (1996) 79–131. Y. Lou, W.M. Ni, Diffusion vs. cross-diffusion: An elliptic approach, J. Differential Equations 154 (1999) 157–190. R. Peng, Qualitative analysis of steady states to the Sel’kov model, J. Differential Equations 241 (2007) 386–398. L. Nirenberg, Topics in Nonlinear Functional Analysis, American Mathematical Society, Providence, RI, 2001. G.M. Lieberman, Bounds for the steady-state Sel’kov model for arbitrary p in any number of dimensions, SIAM J. Math. Anal. 36 (2005) 1400–1406. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equation of Second Order, in: Classics in Mathematics, Reprint of the 1998 edition., SpringerVerlag, Berlin, 2001. [14] R. Peng, J.P. Shi, M.X. Wang, On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law, Nonlinearity 21 (2008) 1471–1488.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
On steady-state solutions of the Brusselator-type systemI Rui Peng a,b,∗ , Ming Yang c a
Institute of Nonlinear Complex Systems, College of Science, China Three Gorges University, Yichang City, 443002, Hubei Province, PR China
b
School of Science and Technology, University of New England, Armidale, NSW, 2351, Australia
c
Department of Mathematics, Southeast University, Nanjing, 210018, PR China
article
info
Article history: Received 14 October 2008 Accepted 1 December 2008 MSC: 35J55 35B45 35K57 92C40 Keywords: Brusselator-type system Steady state Existence Non-existence Asymptotic behavior
a b s t r a c t In this article, we shall be concerned with the following Brusselator-type system:
−θ ∆u = λ(1 − (b + 1)u + bum v) −∆v = λa2 (u − um v)
in Ω , in Ω ,
under the homogeneous Neumann boundary conditions. This system was recently investigated by M. Ghergu in [Nonlinearity, 21 (2008), 2331–2345]. Here, Ω ⊂ RN (N ≥ 1) is a smooth and bounded domain and a, b, m, λ and θ are positive constants. When m = 2, this system corresponds to the well-known stationary Brusselator model which has received extensive studies analytically as well as numerically. In the present work, we derive some further results for the general system. Our conclusions show that there is no non-constant positive steady state for large a while small a may produce non-constant positive steady states. If 1 ≤ N ≤ 3 and 1 < m < 3, we particularly determine the asymptotic behavior of non-constant positive steady states as a converges to zero, thereby solving an open problem left in Ghergu’ work. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction In this article, after a scaling as in [1,2], we shall be concerned with the following general stationary Brusselator system:
−θ ∆u = λ(1 − (b + 1)u + bum v) −∆v = λa2 (u − um v) ∂ u = ∂v = 0 ∂ν ∂ν
in Ω , in Ω ,
(1.1)
on ∂ Ω .
Here, Ω ⊂ RN is a bounded domain with smooth boundary ∂ Ω , ν is the outward unit normal vector on ∂ Ω . The unknown functions u(x) and v(x) represent the spatial concentrations of two intermediate reactants and are considered to be nonnegative, a and b are fixed concentrations of other components, θ is the diffusion coefficient of the concentration u, and λ is a measure of the size of the domain. Therefore, a, b, θ and λ are always assumed to be positive constants. For more interpretation of the background of this chemical model, the interested reader may further refer to [3–5,1,2,6].
I This work was partially supported by the National Natural Science Foundation of China (10801090, 10871185, 10726016, 10601011), the Scientific Research Projects of Hubei Provincial Department of Education (Q200713001) and Scientific Innovation Team Project of Hubei Provincial Department of Education (T200809). ∗ Corresponding address: Department of Mathematics, College of Science, China Three Gorges University, Yichang City, 443002, Hubei Province, PR China. E-mail addresses: [email protected] (R. Peng), [email protected] (M. Yang).
0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.12.003
1390
R. Peng, M. Yang / Nonlinear Analysis 71 (2009) 1389–1394
When m = 2, (1.1) becomes the well-known Brusselator model, which was proposed by Prigogine and Lefever [7] in 1968 to model morphogenesis and pattern formation in chemical reactions. In recent decades, the Brusselator system has received both intensive analytical and numerical studies by many authors, for example, [3–5,2] and the references therein. The fascinating Turing instability and Turing pattern have been observed through this looking-simple coupled elliptic system. Moreover, the associated dynamics of the Brusselator model was dealt with by You in [6]. For the extended form of Brusselator model (1.1), Ghergu [1] also focused on the steady-state problem. His results exhibit the critical role the nonlinear term um plays in generating the non-constant positive solutions (i.e., stationary pattern) of (1.1). To be precise, he found that, if m ≤ 1, (1.1) admits no non-constant positive solution; on the contrary, if m > 1, among other proper conditions on a, b, θ , λ and Ω , (1.1) possesses at least one non-constant positive solution. He also obtained various existence and non-existence conclusions with respect to the parameters b, θ , λ. To achieve these results, in [1], Ghergu first applied the maximum principle (due to [8]) and integration estimates to establish upper and lower bounds for any classical positive solution to (1.1). Then, with the helps of various mathematical tools such as the energy integration, implicit function theorem and topological degree, the work of Ghergu provides lots of interesting criteria for the existence or non-existence for non-constant positive solutions of (1.1), some of which turn out to be the improvement of the results derived by Peng and Wang in [2]. It is clear that only nonnegative solutions of (1.1) are of physical interests. As in the existing works [3–5,1,2], we are also interested in the existence and non-existence of non-constant positive solution to (1.1). From now on, whenever mentioning the positive solution (u, v) of (1.1), we always mean that (u, v) ∈ C 2 (Ω ) × C 2 (Ω ) such that u, v are positive on Ω and satisfy (1.1). Additionally, it should be noted that (u, v) = (1, 1) is the unique constant positive solution of (1.1). For sake of the reader’s reference, we are ready to list the main results of [1] as follows. First of all, by the maximum principle for elliptic equations which comes from Lou and Ni (see, [8], Proposition 2.2 or [9] Lemma 2.1), without any restriction on spatial dimensions, Ghergu established the upper and lower bounds for arbitrary positive solution of (1.1). Theorem 1.1 (See Ghergu [1], Theorem 2.2). Let m > 1, then any positive solution (u, v) of (1.1) satisfies 1 b+1
≤u≤1+
b(b + 1)m−1
θ a2
and
1+
b(b + 1)m−1
θ a2
1−m
≤ v ≤ (b + 1)m−1 .
Then, based on Theorem 1.1 and some other a priori estimates, using several analytical techniques including the maximum principle, energy integration and the method of implicit function theorem (due to Peng and Wang [2] and Peng [10]), Ghergu deduced the following results concerning the non-existence of non-constant positive solutions to (1.1). Theorem 1.2 (See Ghergu [1], Section 3). The system (1.1) has no non-constant positive solution if one of the following cases happens: (i) 0 < m ≤ 1; (ii) λ is small enough; (iii) θ is large enough; (iv) b is small enough. Let 0 = µ0 < µ1 < µ2 < · · · be the eigenvalues of −∆ in Ω with the homogeneous Neumann boundary conditions, and let ei be the algebraic multiplicity of µi . Thus, combining Theorems 1.1 and 1.2, and the calculation formula of Leray–Schauder degree and its invariance with respect to homotopy (see [11]), the author used the argument similar to Theorem 5.1 of [2] to establish some existence results regarding non-constant positive solutions to (1.1). In particular, he showed Theorem 1.3 (See Ghergu [1], Corollary 4.3). Let λ, θ > 0 be fixed and b(m − 1) > 1. Assume that for some i ≥ 1,
λθ −1 (b(m − 1) − 1) ∈ (µi , µi+1 ) and
i X
ek is odd,
(1.2)
k=1
then there exists a A > 0 such that (1.1) has at least one non-constant positive solution for all 0 < a < A. The main purpose of the present work is to derive some further results for (1.1). Firstly, we shall prove the non-existence for non-constant positive solutions to (1.1) in two cases: (1) m is larger than but close to 1; (2) a is sufficiently large. To this end, we apply the implicit function theorem and topological degree argument. More precisely, our first result can be summarized as follows. Theorem 1.4. (i) Let a, b, θ and λ be fixed, then there exists a small 0 > 0 such that (1.1) has no non-constant positive solution if m < 1 + 0 ; (ii) Let b, m, θ and λ be fixed, then there exists a large a > 0 such that (1.1) has no non-constant positive solution if a > a. As the second result, we attempt to determine the asymptotic profile of the non-constant positive solutions obtained in Theorem 1.3 as a converges to zero. For emphasizing the dependence of these non-constant positive solutions (u, v) on a, we denote them to be (ua , va ). It was conjectured in the concluding section of [1] that, as a goes to zero, va approaches a constant profile while the limit of ua satisfies a nonlinear elliptic equation with the homogeneous Neumann boundary condition. Our
R. Peng, M. Yang / Nonlinear Analysis 71 (2009) 1389–1394
1391
statement will give an affirmative answer to this open problem. The key point of our analysis is to establish more delicate a priori estimates for positive solutions to (1.1) as a → 0. However, due to mathematical difficulties, in proving the desired assertion we have to restrict 1 ≤ N ≤ 3; and if N = 3, m also satisfies 1 < m < 3. Taking into consideration the realistic situation of chemistry, the restriction 1 ≤ N ≤ 3 is enough for the possible application of the extended Brusselator system (especially for the Brusselator system with m = 2). Theorem 1.5. Let λ, θ > 0 be fixed, and b(m − 1) > 1 and 1 ≤ N ≤ 3. If N = 3, the constant m also satisfies 1 < m < 3. Assume that for some i ≥ 1, the condition (1.2) is also satisfied. Then, for any sequence of non-constant positive solution (ua , va ) to (1.1), there is a subsequence (uan , van ) of (ua , va ) satisfying that (uan , van ) converges to (u∗ , v ∗ ) on C 2 (Ω )× C 2 (Ω ) as an → 0+ , where u∗ is a non-constant positive solution of
" − θ ∆u = λ 1 − (b + 1)u + bu
m
Z
m
#
−1 Z
u dx
udx
Ω
Ω
in Ω ,
∂u =0 ∂ν
on ∂ Ω ,
(1.3)
and v ∗ is uniquely determined by
v∗ =
Z Ω
(u∗ )m dx
−1 Z Ω
u∗ dx.
Therefore, if (1.2) holds, the non-local elliptic problem (1.3) admits at least one non-constant positive solution. We would like to point out that, in general, it is very hard to prove the existence of non-constant positive solutions of the non-local elliptic equation such as (1.3). In terms of chemistry, Theorem 1.5 implies that, when the concentration of a certain component (represented by a) is small enough, then in the two intermediate reactants, the component v must be spatially homogeneous while the other component u may be spatially heterogeneous. 2. Proof of main results In this section, we shall give the detailed proof of Theorems 1.4 and 1.5. From now on, all the integrals are computed over
Ω . For any 1 ≤ p, we use k · kp and k · k2,p to stand for the usual norms of Banach spaces Lp (Ω ) and W 2,p (Ω ), respectively. As some preliminaries for the proof of Theorem 1.5, we need to recall two basic results. Assume that (u, v) is a nonnegative solution to (1.1), then integrating the equations in (1.1), we find
(b + 1)
Z
Z u−b
um v = |Ω | and
Z
Z u=
um v.
(2.1)
Here and in what follows, |Ω | always represents the volume of the domain Ω . Hence, it follows from (2.1) that
Z
u = |Ω |.
(2.2)
We will have to present a local result for weak super-solution of linear elliptic equations from [12] (also see, for example, [13], Theorem 8.18). Lemma 2.1. Let Ω be a bounded Lipschitz domain in RN . Let Λ be a non-negative constant and suppose that z ∈ W 1,2 (Ω ) is a non-negative weak solution of the inequalities 0 ≤ −∆z + Λz
in Ω ,
∂z = 0 on ∂ Ω . ∂ν
Then, for any q ∈ [1, N /(N − 2)), there exists a positive constant C0 , depending only on q, Λ and Ω , such that
kz kq ≤ C0 inf z . Ω
2.1. Proof of Theorem 1.4 Proof of (ii) of Theorem 1.4. Assume that (ua , va ) is an arbitrary positive solution of (1.1). Then, we first claim that (ua , va ) → (1, 1) uniformly on Ω as a → ∞. By Theorem 1.1, we know both ua and va are uniformly bounded on Ω for all large a. Then, from the standard regularity theory for elliptic equations and imbedding theorems, it follows from the first equation of (1.1) that there is a sequence an with an → ∞ as n → ∞ and the corresponding solution (un , vn ) =: (uan , van ), such that un → u˜ in C 1 (Ω ) as n → ∞. Furthermore, u˜ is a positive function on Ω . Since vn satisfies
−∆vn = λa2n (un − um n vn ) in Ω ,
∂vn = 0 on ∂ Ω , ∂ν
1392
R. Peng, M. Yang / Nonlinear Analysis 71 (2009) 1389–1394
we can proceed as in the proof of Lemma 3.4 of [14] to assert vn → (˜u)1−m uniformly on Ω as a → ∞. As a result, it is easy to see that u˜ is a solution of the following equation
−θ ∆u˜ = λ(1 − u˜ ) in Ω ,
∂ u˜ = 0 on ∂ Ω . ∂ν
This equation has a unique solution u˜ = 1. Therefore, our previous claim holds true. In order to achieve the desired conclusion, we need to use a different argument from those in [1,2] which seem not to work in our case. The argument here is essentially similar to that of the proof for Theorem 3.3 in [14]. We shall mainly apply the theory of topological degree. For this purpose, we shall have to reformulate the system (1.1) in the framework that the Leray–Schauder degree theory can work. Let us denote
Ξ = {(u, v) ∈ C (Ω ) × C (Ω ) : C < u, v < C }, where the positive numbers C and C satisfying 0 < C < 1 and 1 < C will be determined later. We also define the operator
Γ (a, b, u, v) = (−∆ + I)−1 u + θ −1 λ(1 − (b + 1)u + bum v), v + λa2 (u − um v) , where (−∆ + I)−1 stands for the inverse operator of −∆ + I subject to Neumann boundary condition over ∂ Ω . Let a < a and b0 be arbitrary positive constants. It is well-known that Γ is a compact operator from [a, a] × [0, b0 ] × Ξ to C (Ω ) × C (Ω ). Furthermore, (u, v) ∈ Ξ solves (1.1) if and only if (u, v) satisfies (u, v) = Γ (a, b, u, v). In addition, by the fact that (ua , va ) → (1, 1) uniformly on Ω as a → ∞, together with Theorem 1.1, we can find the positive numbers a, C and C , such that
(u, v) 6= Γ (a, b, u, v),
∀a ∈ [a, ∞), b ∈ [0, b0 ] and (u, v) ∈ ∂ Ξ .
As a result, the topological degree deg (I − Γ (a, b, ·), Ξ , 0) is well-defined and is also independent of a ∈ [a, ∞) and b ∈ [0, b0 ]. We recall that Theorem 1.1 with b = 0 implies that (1, 1) is the unique fixed point of Γ (a, 0, ·) in Ξ , and thus deg(I − Γ (a, 0, ·), Ξ , 0) = index (I − Γ (a, 0, ·), (1, 1)). On the other hand, the remark of Section 5 in [1] states that (1, 1) is linearly stable as the unique constant positive steady state for the corresponding reaction–diffusion system of (1.1) with (a, b) = (a, 0) if we take a to be larger if necessary. Hence, by the well-known Leray–Schauder degree formula (see, e.g., Theorem 2.8.1 in [11]), we have deg(I − Γ (a, 0, ·), Ξ , 0) = index (I − Γ (a, 0, ·), (1, 1)) = 1. Therefore, it follows that deg(I − Γ (a, b, ·), Ξ , 0) = deg (I − Γ (a, 0, ·), Ξ , 0) = 1, for any a ∈ [a, ∞) and b ∈ [0, b0 ]. We have shown that for any positive solution (ua , va ) of (1.1), (ua , va ) → (1, 1) uniformly on Ω as a → ∞. Therefore, according to the discussion of Section 5 in [1], carrying out the same elementary computation as in Theorem 2.1 of [2], we can take a larger a if necessary such that every possible positive solution (ua , va ) of (1.1) is linearly stable provided that a ≥ a. Hence, the fixed point index of Γ (a, b, ua , va ) is well-defined and is equal to 1 if a ≥ a and b0 ≥ b ≥ 0. Furthermore, for such fixed a and b, the compactness of Γ (a, b, ·) implies that there are at most finitely many such fixed points in Ξ , denoted by {(ui , vi )}`1 . Then, from the additivity property of the Leray–Schauder degree, it follows that 1 = deg(I − Γ (a, b, ·), Ξ , 0) =
` X
index (I − Γ (a, b, ·), (ui , vi )) = `.
1
This indicates the uniqueness of positive solutions of (1.1) for a ≥ a and so the unique positive solution must be (1, 1). Our proof is complete. The proof of (i) in Theorem 1.4 can be obtained similarly to the above but is simpler. It is sufficient to note that, for m = 1, (1.1) has a unique positive solution (1, 1) which is also linearly stable when m is close to 1. We leave the details to the interested reader. 2.2. Proof of Theorem 1.5 Proof. For sake of clarity, we divide our proof into two steps. Step 1. We first have to establish some a priori upper and lower estimates for positive solutions (ua , va ) of (1.1) as a → 0+ . In virtue of Theorem 1.1, it remains to derive the upper bound of ua and lower bound of va .
R. Peng, M. Yang / Nonlinear Analysis 71 (2009) 1389–1394
1393
We first show that kua k∞ is bounded from above as a → 0+ . Note that
∂ ua = 0 on ∂ Ω . ∂ν
−θ ∆ua + λ(b + 1)ua > 0 in Ω , Thus, by use of Lemma 2.1, we have
kua kq ≤ C0 inf ua .
(2.3)
Ω
Here, q ≥ 1 can take arbitrary large number if N = 1 or 2; and q ∈ [1, 3) if N = 3. Here and in what follows, the positive constant C0 depends only on b, m, λ, θ and Ω , and may vary from place to place. From (2.3), it is not hard to see that
k ua k q ≤ C 0 .
(2.4)
Otherwise, we can find a sequence {an } with an → 0 as n → ∞ and a corresponding positive solution sequence {(uan , van )}, such that kuan kq → ∞. Thus, from (2.3), it follows uan → ∞ uniformly on Ω as n → ∞. This is a contradiction with (2.2). We also observe that va ≤ (b + 1)m−1 on Ω . When N = 1 or 2, by taking q = 2m in (2.4), we find +
kλ(1 − (b + 1)ua + bum a va )k2 ≤ C0 . Then, in accordance with the equation of ua , the well-known Lp theory for elliptic equations gives kua k2,2 ≤ C0 . Hence, for N = 1, 2, the imbedding theorem W 2,2 (Ω ) ,→ C α (Ω ) for some α ∈ (0, 1) concludes kua k∞ ≤ C0 , as required. If N = 3, together with (2.4) for 1 ≤ q < 3, we obtain the following estimates:
kλ(1 − (b + 1)ua + bum a va )kp ≤ C0 , for 1 ≤ p <
3 m
.
Thanks to the Lp theory for elliptic equations again, one gets kua k2,p ≤ C0 . When 3 − 2p ≤ 0, the imbedding theorem W 2,p (Ω ) ,→ Lr (Ω ) for all 1 ≤ r < ∞ deduces kua kr0 ≤ C0 for some r0 > 3/2. Thus, using the equation of ua and the imbedding theorem W 2,r0 (Ω ) ,→ C α (Ω ) for some α ∈ (0, 1) again, we can easily obtain kua k∞ ≤ C0 . When 3 − 2p > 0, it follows from W 2,p (Ω ) ,→ Lr1 (Ω ) for any 1 ≤ r1 < 3p/(3 − 2p) that kua kr1 ≤ C0 for 1 ≤ r1 < 3p/(3 − 2p). Once again, we obtain kua k2,r1 ≤ C0 and so kua kr2 ≤ C0 for 1 ≤ r2 < 3r1 /(3 − 2r1 ). The condition 1 < m < 3 allows us to repeat the bootstrapping procedure so that we find a rn such that 3 − 2rn ≤ 0. In this case, we can continue to use the analysis of the above paragraph to assert kua k∞ ≤ C0 . Next, we verify the fact that there exists a positive lower bound for va for all small a. If this is false, we can find a sequence {an } with an → 0+ as n → ∞ and a corresponding positive solution sequence {(un , vn )}, such that infΩ vn → 0 as n → ∞. Combining the standard regularity theory for elliptic equations and imbedding theorems and the equations of un and vn , passing to a further sequence of {an } if necessary, we may assume that un → u∗ and vn → v ∗ in C 2 (Ω ), and u∗ > 0 on Ω due to Theorem 1.1. In addition, v ∗ ≥ 0 on Ω and satisfies
∂v ∗ = 0 on ∂ Ω , ∂ν
−∆v ∗ = 0 in Ω ,
which implies v ∗ ≡ 0 on Ω . On the other hand, by (2.1), we notice
Z lim
n→∞
Z un =
u∗ =
Z
(u∗ )m v ∗ = 0,
and so this arrives at a contradiction with (2.2). Until now, we have verified that there exist two positive numbers C1 and C2 which is independent of all small a, such that for any positive solution (ua , va ) of (1.1), the following holds: C1 ≤ ua (x), va (x) ≤ C2 ,
∀x ∈ Ω .
(2.5)
Step 2. Based on the estimates (2.5), we prove the asymptotic behavior of non-constant positive solutions obtained in Theorem 1.3. Suppose that our conclusion stated in Theorem 1.5 is not true. Then, using (2.5), the standard Lp and Schauder’s estimates and the imbedding theorems, there exists a sequence {an } with an → 0+ as n → ∞ and the corresponding nonconstant positive solution (un , vn ) to (1.1) for a = an , such that there is a subsequence of (un , vn ), still denoted by itself, ¯ ) × C 2 (Ω ¯ ) as n → ∞. satisfying (un , vn ) → (1, 1) on C 2 (Ω Since (un , vn ) is non-constant in Ω for each n ≥ 1, we can define hn = un − 1,
kn = vn − 1,
and h˜ n =
hn
khn k∞ + k k n k∞
,
k˜ n =
kn
khn k∞ + kkn k∞
.
1394
R. Peng, M. Yang / Nonlinear Analysis 71 (2009) 1389–1394
As a result, h˜ n and k˜ n satisfy
˜ −θ ∆h˜ n = −λ(b + 1)h˜ n + bmλζnm−1 h˜ n + bλum n kn 2˜ 2 m−1 ˜ 2 m˜ ˜ −∆kn = λan hn − mλan ζn hn − λan un kn ˜ ˜ ∂ hn = ∂ kn = 0 ∂ν ∂ν
in Ω , in Ω ,
(2.6)
on ∂ Ω ,
where ζn (x) lies between un (x) and 1. Applying the standard theory, up to a further sequence, we can assume that (h˜ n , k˜ n ) → ¯ ) × C 1 (Ω ¯ ) as n → ∞. By passing to the limit in (2.6), we find that (h˜ , k˜ ) in C 1 (Ω
−θ ∆h˜ = λ(bm − (b + 1))h˜ + bλk˜ ,
−∆k˜ = 0 in Ω ,
∂ h˜ ∂ k˜ = = 0 on ∂ Ω . ∂ν ∂ν
Hence, k˜ is a constant and so our hypothesis λθ −1 (bm − (b + 1)) 6= µn for any n ≥ 0 deduces
(bm − (b + 1))h˜ + bk˜ = 0.
(2.7)
Furthermore, kh˜ n k∞ + kk˜ n k∞ = 1 for each n ≥ 1 gives
|h˜ | + |k˜ | = 1. On the other hand, integrating the second equation in (2.6), dividing the resulting integral by
(2.8) a2n
and then letting n → ∞,
˜ which, together with (2.7) and (2.8), leads to a contradiction. Consequently, under the assumption we obtain (m − 1)h˜ = k, of Theorem 1.5, simple analysis asserts, for any sequence of non-constant positive solution (ua , va ) to (1.1), there is a subsequence (uan , van ) of (ua , va ) satisfying that (uan , van ) converges to (u∗ , v ∗ ) on C 2 (Ω ) × C 2 (Ω ) as an → 0+ , where u∗ and v ∗ are determined as in Theorem 1.5. Hence, the proof is now finished. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
M. Ghergu, Non-constant steady-state solutions for Brusselator type systems, Nonlinearity 21 (2008) 2331–2345. R. Peng, M.X. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl. 309 (2005) 151–166. K.J. Brown, F.A. Davidson, Global bifurcation in the Brusselator system, Nonlinear Anal. TMA 12 (1995) 1713–1725. T.K. Callahan, E. Knobloch, Pattern formation in three-dimensional reaction–diffusion systems, Physica D 132 (1999) 339–362. T. Erneux, E. Reiss, Brusselator isolas, SIAM J. Appl. Math. 43 (1983) 1240–1246. Y. You, Global dynamics of the Brusselator equations, Dyn. Partial Diff. Eqns. 4 (2007) 167–196. I. Prigogene, R. Lefever, Symmetry breaking instabilities in dissipative systems II, J. Chem. Phys. 48 (1968) 1665–1700. Y. Lou, W.M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations 131 (1996) 79–131. Y. Lou, W.M. Ni, Diffusion vs. cross-diffusion: An elliptic approach, J. Differential Equations 154 (1999) 157–190. R. Peng, Qualitative analysis of steady states to the Sel’kov model, J. Differential Equations 241 (2007) 386–398. L. Nirenberg, Topics in Nonlinear Functional Analysis, American Mathematical Society, Providence, RI, 2001. G.M. Lieberman, Bounds for the steady-state Sel’kov model for arbitrary p in any number of dimensions, SIAM J. Math. Anal. 36 (2005) 1400–1406. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equation of Second Order, in: Classics in Mathematics, Reprint of the 1998 edition., SpringerVerlag, Berlin, 2001. [14] R. Peng, J.P. Shi, M.X. Wang, On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law, Nonlinearity 21 (2008) 1471–1488.