Abstract
In this article, we make bifurcation analysis on the FKPP type equation under periodic boundary condition. And we show that the solutions bifurcate from the trivial solution u = 0 to an attractor ∑_{λ }as parameter crosses certain critical value. Moreover, we prove that the attractor ∑_{λ }consists of only one cycle of steady state solutions and is homeomorphic to S^{1}. The analysis is based on a new theory of bifurcation, called attractor bifurcation, which was developed by Ma and Wang.
2000 Mathematics Subject Classification: 35B; 35Q; 37G; 37L.
Keywords:
FKPP type equation; periodic boundary condition; attractor bifurcation; center manifold1 Introduction
We consider the following reaction diffusion equation of FKPP type:
supplemented with the following natural constraint:
where a > 0, d > 0, b are given and λ > 0 is system parameter.
In 1937, the FKPP equation was first proposed by Fisher as a model to describe the propagation of advantageous genes [1] and was studied mathematically by Kolmogorov et al. [2]. Moreover, it was also used as biological models for population dynamics [35].
The FKPP equation has been extensively studied during the last decades. Among all the topics of these, the existence of traveling waves (exact form of solutions) and the asymptotic behavior of solutions attract much attention. Many different kinds of methods for the existence of traveling waves (exact form of solutions) haven been developed, such as Painleve expansion method [6,7], bilinear method [8,9], symmetry methods [10]. On the other hand, many results on the asymptotic behavior of solutions are also obtained; see among others [1114] and references therein.
However, there is few work on the attractor bifurcation of the FKPP equation. As a new notion of bifurcation, attractor bifurcation was developed by Ma and Wang [1517] and attracted researchers [18,19]. Ma and Wang [15] first proposed this new notion and applied it to RayleighBenard Convection. Afterwards, with this new theory, Park [18] analyzed the bifurcation of the complex GinzburgLandau equation (CGLE) and Zhang et al. [19] studied the attractor bifurcation of the KuramotoSivashinsky equation.
In this article, we focus on the attractor bifurcation of FKPP type Equation (1.1). The bifurcation analysis near the first eigenvalue of (1.1) will be discussed. The topology structure of the bifurcated solutions will also be studied. As a result, we show the system bifurcates from the trivial solution to an attractor ∑_{λ }as system parameter λ crosses the critical value a, the first eigenvalue of the eigenvalue problem of the linearized equation of (1.1). Furthermore, we prove that ∑_{λ }is homeomorphic to S^{1 }and consists of only one cycle of steady state solutions.
This article is organized as follows. The mathematical setting are given in Section 2. The main results are stated in Section 3. The preliminaries are put in section 4. And Section 5 devote to the proof of the main theorem.
2 Mathematical setting
Let
and we define L_{λ }= A + B_{λ }: H_{1 }→ H and G : H_{1 }→ H by
Consequently, we have an operator equation which is equivalent to the problem (1.1):
Thanks to the existence result of semilinear evolution equations, see Temam [20], Pazy [21], we can define a semigroup
which satisfies the semigroup property.
3 Main results
3.1 The definition of attractor bifurcation
In order to state the main theorem of this article, we start with the definition of attractor bifurcation which was first proposed by Ma and Wang in [1517].
Let H and H_{1 }be two Hilbert spaces, and H_{1 }↪ H be a dense and compact inclusion. We consider the following nonlinear evolution equations
where u : [0, ∞) → H is the unknown function, λ ∈ R is the system parameter, and L_{λ}:H_{1}: → H are parameterized linear completely continuous fields depending continuously on λ, which satisfy
Since L_{λ }is a sectorial operator which generates an analytic semigroup for any λ ∈ R, we can define fractional power operators (L_{λ})^{μ }for 0 ≤ μ ≤ 1 with domain H_{μ }= D((L_{λ})^{μ}) such that if μ_{1 }> μ_{2}, and H_{0 }= H. In addition, we assume that the nonlinear terms G(., λ) : H_{α }→ H for some 0 ≤ α < 1 are a family of parameterized C^{r }bounded operator (r ≥ 1) continuously depending on λ, such that
Definition 3.1.1 A set Σ ⊂ H is called an invariant set of (3.1) if S(t) Σ = Σ for any t ≥ 0. An invariant set Σ ⊂ H of (3.1) is said to be an attractor if Σ is compact, and there exists a neighborhood of W ⊂ H of Σ such that for any ψ_{0 }∈ W we have
Definition 3.1.2 (1) We say that the solution to Equation (3.1) bifurcates from (ψ, λ) = (0, λ_{0}) to an invariant set Ω_{λ}, if there exists a sequence of invariant sets of (3.1) such that , and
(2) If the invariant sets Ω_{λ }are attractors of (3.1), then the bifurcation is called attractor bifurcation.
3.2 Main theorem
In this article, based on attractor bifurcation theory we obtain the following results.
Theorem 3.2.1 For the problem (1.1) with (1.2), if 2b^{2 }< 9ad, following assertions hold true:
(1) if λ ≤ a, the steady state u = 0 is locally asymptotically stable. Furthermore, if b = 0, the steady state u = 0 is globally asymptotically stable.
(2) if λ > a, the Equation (1.1) bifurcates from u = 0 to an attractor Σ_{λ }which is homeomorphic to S^{1}.
(3) Σ_{λ }consists of exactly one cycle of steady solutions of (1.1).
(4) There exists a neighborhood U of u = 0, such that Σ_{λ }attracts U/Γ, where Γ is the stable manifold of u = 0 with codimension 2 in H.
4 Preliminaries
4.1 Attractor bifurcation theory
In the following, we proceed with the principle of exchange of stabilities (PES). Let the eigenvalues (counting multiplicity) of L_{λ }be given by
Suppose that
Let the eigenspace of L_{λ }at λ = λ_{0 }be
It is known that dim E_{0 }= m.
Theorem 4.1.1[17] Assume that the conditions (3.2), (3.3), (4.1), and (4.2) hold true, and u = 0 is locally asymptotically stable for (3.1) at λ = λ_{0}. Then the following assertions hold true:
(1) (3.1) bifurcates from (u, λ) = (0, λ_{0}) to attractors Ω_{λ}, having the same homology as S^{m}^{1}, for λ > λ_{0}, with m  1 ≤ dim Ω_{λ }≤ m, which is connected as m > 1;
(2) for any u_{λ }∈ Ω_{λ}, u_{λ }can be expressed as
(3) There is an open set U ⊂ H with 0 ∈ U such that the attractor Ω_{λ }bifurcated from (0, λ_{0}) attracts U/Γ in H, where Γ is the stable manifold of u = 0 with codimension m.
To get the structure of the bifurcated solutions, we introduce another theorem.
Let v be a twodimensional C^{r }(r ≥ 1) vector field given by
for x ∈ R^{2}. Here
where F_{k }is a kmultilinear field, which satisfies an inequality
for some constants 0 < C_{1 }< C_{2 }and k = 2m + 1, m ≥ 1.
Theorem 4.1.2 ([[16], Theorem 5.10, p. 134]) Under conditions (4.4) and (4.5), the vector field (4.3) bifurcates from (x, λ) = (0, 0) to an attractor Π_{λ }for λ > 0, which is homeomorphic to S^{1}. Moreover, one and only one of the following is true.
(1) Π_{λ }is a period orbit,
(2) Π_{λ }consists of infinitely many singular points,
(3) Π_{λ }contains at most 2(k + 1) = 4(m + 1) singular points, and has 4N + n (N + n ≥ 1) singular points, 2N of which are saddle points, 2N of which are stable node points (possibly degenerate), and n of which have index zero.
4.2 Center manifold reduction
Since the key point in the proof of Theorem 3.2.1 is the center manifold function, we introduce an approximation formula of the center manifold function derived in [16].
We assume that the spaces H_{1 }and H can be decomposed into
where are invariant spaces of L_{λ}, i.e., L_{λ }can be decomposed into such that for any λ near λ_{0},
where all eigenvalues of possess negative real parts, all eigenvalues of possess nonnegative real parts at λ = λ_{0}.
Thus, for λ near λ_{0}, (3.1) can be rewritten as
where , and are canonical projections. Moreover, let
where α < 1 given by (3.3).
By the classical center manifold theorem (see among others [20,22]), there exists a neighborhood of λ_{0}, a neighborhood of x = 0, and a C^{1 }center manifold function , depending continuously on λ. Then to investigate the dynamic bifurcation of (3.1) it suffices to consider the finite dimensional system as follows
Assume the nonlinear operator G be in the following form
for some integer k ≥ 2. Here G_{k }is a kmultilinear operator
Theorem 4.2.1 ([[16], Theorem 3.8, p. 63]) Under the conditions (3.2), (4.1), (4.2), (4.6), and (4.8), the center manifold function Φ(x, λ) can be expressed as
where is as in (4.7), the canonical projection, , and the eigenvalues of .
5 Proof of main theorem
In this section, we shall prove Theorem 3.2.1 by four steps.
Step 1. In this step, we shall study the eigenvalue problem of the linearized equation of (2.1) and shall find the eigenvectors and the critical value of λ.
Consider the eigenvalue problem of the linear equation,
Note that the eigenvalues and eigenvectors of Laplace operator, which satisfy
are
Then the eigenvalues and eigenvectors of (5.1) are
Now, we get the PES:
As a result, conditions (4.1) and (4.2) are verified.
Step 2. We verify that for any λ ∈ R, operator L_{λ }+ G satisfies conditions (3.2) and (3.3).
From the theory of elliptic equations, operator A : H_{1 }→ H is a homeomorphism. Note that conclusion H_{1 }↪ H is compact, then operator B_{λ }: H_{1 }→ H is linear compact operator. Thanks to the results of analytic semigroup in [2022], from (5.2) we know that operator L_{λ }: H_{1 }→ H is a sectorial operator which generates an analytical semigroup. Condition (3.2) is verified.
It is easy to get the following inequality
condition (3.3) is verified.
Step 3. In this part, we shall prove the existence of attractor bifurcation and analyze the topological structure of attractor Σ_{λ}.
Let . Let Φ be the center manifold function, in the neighborhood of (u, λ) = (0, a), we have
where y = x_{1}e_{1 }+ x_{2}e_{2}.
Then the reduction equations of (2.1) are as follows
To get the exact form of the reduction equations, we need to obtain the expression of < G(u), e_{1 }> and < G(u), e_{2 }>.
Let G_{2 }: H_{1 }× H_{1 }→ H and G_{3 }: H_{1 }× H_{1 }× H_{1 }→ H are the bilinear and trilinear operators of G, respectively, i.e.,
Since
the firstorder approximation of (5.3) doesn't work. Now, we shall find out the secondorder approximation of (5.3). And the most important of all is to obtain the approximation expression of the center manifold function.
Since
and
we can obtain
By the formula (4.9) in Theorem 4.2.1 and (5.7), the center manifold function Φ, in the neighborhood of (u, λ) = (0, a), can be expressed as
In the following, we calculate < G(u), e_{j }>, j = 1, 2.
Next we only need to find out < G_{2}(y, Φ(y)), e_{j }>, < G_{2}(Φ(y),y), e_{j }> and G_{3}(y, y, y), e_{j }>.
By calculation, we have
Note that (5.4) hold true, we have
Since
we have
then we obtain the expression of < G(u), e_{j }>, j = 1,2.
Put (5.8) into (5.3), we have the approximation of reduction equation
For the case of λ < a, it is obviously that u = 0 is locally asymptotically stable. For the case of λ = a, if 9ad > 2b^{2}, which implies that A < 0, then u = 0 is also locally asymptotically stable. In particular, if b = 0, λ ≤ a, by Poincaré inequality we have
which implies that u = 0 is globally asymptotically stable. Assertion (1) of Theorem 3.2.1 be proved.
Since the following equality holds true
according to Theorem 4.1.2, we can conclude that if λ > a, the equation (1.1) bifurcates from u = 0 to an attractor Σ_{λ }which is homeomorphic to S^{1}. Assertion (2) and (4) of Theorem 3.2.1 are proved.
Step 4. In the last step, we shall show that the bifurcated attractor of (2.1) contains a singularity cycle.
By Krasnoselskii Theorem for potential operator, at least, L_{λ }+ G bifurcates from (u, λ) = (0,a) to a steady solutions (V_{λ}, λ).
Since the solutions of (2.1) are translation invariant,
the set
represents S^{1 }in H_{1}, which implies that assertion (3) of Theorem 3.2.1 is proved.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author was very grateful to the anonymous referees whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. This research was supported by Scientific Research Foundation of Civil Aviation Flight University of China (J201130).
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