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Mathematical Problems in Engineering
Volume 2010, Article ID 529869, 18 pages
http://dx.doi.org/10.1155/2010/529869
Research Article

Hopf Bifurcation with the Spatial Average of an Activator in a Radially Symmetric Free Boundary Problem

Department of Mathematics, Kyonggi University, Suwon 443-760, Republic of Korea

Received 5 August 2010; Accepted 21 October 2010

Academic Editor: Oded Gottlieb

Copyright © 2010 YoonMee Ham. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An interface problem derived by a bistable reaction-diffusion system with the spatial average of an activator is studied on an -dimensional ball. We analyze the existence of the radially symmetric solutions and the occurrence of Hopf bifurcation as a parameter varies in two and three-dimensional spaces.

1. Introduction

The study of interfacial patterns is important in several areas of biology, chemistry, physics, and other fields [14]. Internal layers (or free boundary), which separate two stable bulk states by a sharp transition near interfaces, are often observed in bistable reaction-diffusion equations when the reaction rate is faster than the diffusion effect. We consider a reaction-diffusion system with a sufficiently small positive constant [5, 6]

where , , and are positive constants, is the ball in -dimensional space, and stands for the unit outward normal on the boundary . The nonlinear functions are

where and denotes the spatial average, describing a global feedback effect, namely,

The system (1.1) with (1.2) is a model for flow discharges proposed by et al. [7], in which is interpreted by the current density and by the voltage drop across the gas gap [7, 8] in a gas-discharge system. This system also exhibits a codimension-two Turing Hopf bifurcation [9], where the conditions of a spatial Turing instability [10] with a certain wavelength and a temporal Hopf bifurcation with a certain frequency are met simultaneously. Equation (1.1) determines the dynamics of an internal layer, and equation (1.1) together with (1.2) represents a basic model of globally coupled bistable medium which is relevant for current density dynamics in large area bistable semiconductor systems [1114]. The internal layer has a physical reason as the current filament has a sharp profile with a narrow transition layer connecting flat on- and off-states.

When in (1.1) is sufficiently small for the case of without the spatial average, the singular limit analysis is applied to show the existence and the stability of localized radially symmetric equilibrium solutions [15, 16]. In one-dimensional space for the case of without the spatial average, such equilibrium solutions should undergo certain instabilities, and the loss of stability resulting from a Hopf bifurcation produces a kind of periodic oscillation in the location of the internal layers [2, 1719]. As the parameter varies, the stability of the spherically symmetric solutions and their symmetry-breaking bifurcations into layer solutions for the case of without the spatial average have been examined in [5, 6].

In this paper, the free boundary problem of (1.1) with (1.2) for the case when in two- and three-dimensional space will be studied. Suppose that there is only one -dimensional hypersurface which is a single closed curve given in the domain in such a way that , where and . When and in (1.1), the spatial average satisfies

where and . The spatial average of is a solution of . Equation of is given by (see [20, 21])

where is the outward normal vector on , is the value of on the interface , and is the velocity of the interface. The reaction terms (1.2) satisfy the bistable condition, that is, the nullclines of and must have three intersection points and the nullcline is the triple-valued function of which is called , , and . From [2, 20, 22], the trajectory with a unique value of exists which is given by . Furthermore, the velocity of the interface is a continuously differentiable function defined on an interval , and thus the velocity of the interface can be normalized by

where .

An analysis of the dynamics of this process has been shown (see, e.g., [2, 5, 6, 15]) to lead a free boundary problem consisting of the initial-boundary value problem

where .

The organization of the paper is as follows. In Section 2, a change of variables is given which regularizes problem (1.7) in such a way that results from the theory of nonlinear evolution equations can be applied. In this way, we obtain enough regularity of the solution for an analysis of the bifurcation. In Section 3, we show the existence of radially symmetric localized equilibrium solutions for (1.7) and obtain the linearization of problem (1.7). In the last section we show the existence of the periodic solutions and the bifurcation of the interface problem as a parameter varies in two and three dimensions.

2. Regularized System

We look for an existence problem of radially symmetric equilibrium solutions of (1.7) with , where the center and the interface are located at the origin and , respectively. The problem is given by

As a first step we obtain more regularity for the solution by semigroup methods, considering as a densely defined operator , where and with norm .

We define ,

and ,

Here is a Green's function of satisfying the Neumann boundary conditions: for ,

and for ,

where and are modified Bessel functions () and is given by

Moreover, for all and , and .

Applying the transformation , then we obtain an equivalent abstract evolution equation of(2.1)

where is a matrix defined on and given by

The nonlinear forcing term defined on the set as

where , and . We define and then . Let , then the velocity of is written by

Lemma 2.1. The functions , and are continuously differentiable with derivatives given by where and .

The well posedness of solutions was shown in [23] applying the semigroup theory using domains of fractional powers of and [24]. Moreover, they obtained that is a continuously differentiable function, where .

3. Radially Symmetric Equilibrium Solutions and Linearization

The steady states are solutions of the following problem:

for .

Lemma 3.1. Define . Then , and for .

Proof. For , the derivative of is given by Let . Then , and . The derivative of is Thus , and thus for . Moreover, Since are increasing functions and are decreasing functions (), and thus for . For , the derivative of is where . Since , we have for . Moreover, for .

Theorem 3.2. Suppose that (i) and or (ii) and . Then the stationary problem of (2.7) has the only stationary solution for all with and ,. The linearization of at is The pair corresponds to a unique steady state of (2.1) for with .

Proof. From the system (3.1), and are solutions of the following equations: We only check the existence of of (2.10) and (3.8), and thus we let Then Since and for ,there is a unique when , , , or , .
The formula for follows from Lemma 2.1, the relation , and . The corresponding steady state for (2.1) is obtained using the transformation and Theorem 2.1 in [17].

Definition 3.3. Suppose that and satisfy and . One defines (for ) the operator that is a linear operator from to as One then defines to be a Hopf point for (2.7) if there exists an and a -curve ( denotes the complexification of the real space ) of eigendata for such that (i), (ii) with ,(iii) for all in the spectrum of , (iv) (transversality), where .

4. Hopf Bifurcation Analysis

We will show that there is a Hopf bifurcation from the curve of radially symmetric stationary solution. The linearized eigenvalue problem of (2.7) is

where is a 3 by 3 identity matrix. This is equivalent to

Our main theorem is stated as follows.

Theorem 4.1. Suppose that and satisfy and , the problem (2.7), and (2.1), has a unique stationary solution , where and , and , respectively, for all . Then there exists a unique such that the linearization has a purely imaginary pair of eigenvalues . The point is then a Hopf point for (2.7), and there exists a -curve of nontrivial periodic orbits for (2.7) and (2.1), bifurcating from and , respectively.

We will show the following three theorems that verify the above theorem. The next theorem shows that the steady state is the only Hopf point.

Theorem 4.2. For , the operator has a unique pair of purely imaginary eigenvalues . Then the point satisfies the conditions (i), (ii), and (iii) in Definition 3.3.

Proof. In the sequel, we denote , . We assume without loss of generality that and is the (normalized) eigenfunction of with eigenvalue . We have to show that can be extended to a -curve of eigendata for with . For this, let . First, we note that if , then (vice versa) in the last equation of (4.2). We see that and , for otherwise, by (4.2), , which is not possible because is symmetric. So without loss of generality, let . Define The equation is equivalent to being an eigenvalue of with eigenfunction . By (4.2), we have which is equivalent to To apply the implicit function theorem to , we have to check that is in and that In addition, the mapping is a compact perturbation of the mapping which is invertible. In order to verify (4.5), it suffices to show that the system which are necessarily implies that , and . We define , then the first equation of (4.9) is given by Since solves (4.2), is a solution to the equation Multiply (4.11) by and integrate, then which implies that Therefore we obtain Multiply (4.10) by and (4.11) by . Now we integrate the resulting equation to obtain
Multiply (4.10) and (4.11) by , and then eliminate the term . Integrating the resulting equation, we have by (4.16) and (4.17). Using (4.9), (4.12), and (4.13) in the above equation, then we obtain Suppose that . Then the real and imaginary parts of (4.19) are given by From these equations, we have This leads to a contradiction that the right hand side is positive for and . Therefore, we should have . Thus and .

Theorem 4.3. Under the same condition as in Definition 3.3, satisfies the transversality condition. Hence this is a Hopf point for (2.7).

Proof. By implicit differentiation of , This means that the functions , and satisfy the equations By letting , then From (4.9) and (4.23), we obtain We now multiply (4.24) by and (4.11) by and then subtract these two equations, then we get Comparing to (4.11) and then integrating, we have Using (4.16) and (4.17) in the above equation, then By substituting (4.12), (4.13), and (4.23), we have The real part of is given by where Since for , we have . Therefore, for and for , and thus by the Hopf-bifurcation theorem in [17], there exists a family of periodic solutions which bifurcates from the stationary solution as passes .

The next theorem shows that a critical Hopf point exists uniquely.

Theorem 4.4. Under the same condition as in Definition 3.3, there exists a unique, purely imaginary eigenvalue of (4.2) with for a unique critical point in order for to be a Hopf point.

Proof. We only need to show that the function has a unique zero with and . This means solving the system (4.2) with , , and , The second equation becomes where is a Green's function of the differential operator . The real and imaginary parts of this above equation are given by Since in [17, Lemma 12] and , there is a unique in the first equation if it does guarantee the existence of . Now, we let then and by assumption. If we show that , then the existence of is proved: Since for and in [17, Lemma 12], we have for all .

There is a unique pure imaginary eigenvalue and the critical point of (2.1) and thus there exists a family of periodic solutions which bifurcates from the stationary solution as passes under the condition of Theorem 4.4. Thus we also found the relationship between and for which Hopf bifurcation occurs for the problem (2.1).

Acknowledgment

This paper was supported by Kyonggi University Grant 2008.

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