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Communications on Pure & Applied Analysis

2004 , Volume 3 , Issue 2

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Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient
P.E. Kloeden
2004, 3(2): 161-173 doi: 10.3934/cpaa.2004.3.161 +[Abstract](38) +[PDF](208.0KB)
The zero solution of a vector valued differential equation with an autonomous linear part and a homogeneous nonlinearity multiplied by an almost periodic function is shown to undergo pitchfork or transcritical bifurcations to small nontrivial almost periodic soutions as a leading simple real eigenvalue of the linear part crosses the imaginary axis.
Multiple positive periodic solutions for a delay host macroparasite model
Shangbing Ai
2004, 3(2): 175-182 doi: 10.3934/cpaa.2004.3.175 +[Abstract](25) +[PDF](183.0KB)
A scalar non-autonomous periodic differential equation with delays arising from a delay host macroparasite model is studied. Two results are presented for the equation to have at least two positive periodic solutions: the hypotheses of the first result involve delays, while the second result holds for arbitrary delays.
A class of generalized symmetries of smooth flows
L. Bakker and  G. Conner
2004, 3(2): 183-195 doi: 10.3934/cpaa.2004.3.183 +[Abstract](139) +[PDF](199.1KB)
A class of generalized space-time symmetries is defined by extending the notions of classical symmetries and reversing symmetries for a smooth flow to arbitrary constant reparameterizations of time. This class is shown to be the group-theoretic normalizer of the abelian group of diffeomorphisms generated by the flow. Also, when the flow is nontrivial, this class is shown to be a nontrivial subgroup of the group of diffeomorphisms of the manifold, and to have a one-dimensional linear representation in which the image of a generalized symmetry is its unique constant reparameterization of time. This group of generalized symmetries and several groups derived from it (among which are the multiplier group and the reversing symmetry group) are shown to be nontrivial but incomplete invariants of the smooth conjugacy class of a smooth flow. Several examples are given throughout to illustrate the theory.
Existence of solutions to equations for the flow of an incompressible fluid with capillary effects
D. L. Denny
2004, 3(2): 197-216 doi: 10.3934/cpaa.2004.3.197 +[Abstract](21) +[PDF](233.9KB)
We study the initial-value problem for a system of equations that models the low-speed flow of an inviscid, incompressible fluid with capillary stress effects. The system includes hyperbolic equations for the density and velocity, and an algebraic equation (the equation of state). We prove the local existence of a unique, classical solution to an initial-value problem with suitable initial data. We also derive a new, a priori estimate for the density, and then use this estimate to show that, if the regularity of the initial data for the velocity alone is increased, then the regularity of the solution for the density and the velocity may be increased, by a bootstrapping argument.
High order product integration methods for a Volterra integral equation with logarithmic singular kernel
T. Diogo , N. B. Franco and  P. Lima
2004, 3(2): 217-235 doi: 10.3934/cpaa.2004.3.217 +[Abstract](40) +[PDF](234.1KB)
This work is concerned with the construction and analysis of high order product integration methods for a class of Volterra integral equations with logarithmic singular kernel. Sufficient conditions for the methods to be convergent are derived and it is shown that optimal convergence orders are attained if the exact solution is sufficiently smooth. The case of non-smooth solutions is dealt with by making suitable transformations so that the new equation possesses smooth solutions. Two particular methods are considered and their convergence proved. A sample of numerical examples is included.
Regularity of the attractor for kp1-Burgers equation: the periodic case
Mostafa Abounouh and  Olivier Goubet
2004, 3(2): 237-252 doi: 10.3934/cpaa.2004.3.237 +[Abstract](25) +[PDF](236.6KB)
We prove the existence of a global attractor for a damped-forced Kadomtsev-Petviashvili equation. We also establish that this equation features an asymptotic smoothing effect. We use energy estimates in conjunction with a suitable splitting of the solutions.
Multiple solutions with changing sign energy to a nonlinear elliptic equation
A. El Hamidi
2004, 3(2): 253-265 doi: 10.3934/cpaa.2004.3.253 +[Abstract](75) +[PDF](222.3KB)
In this paper, the existence of multiple solutions to a nonlinear elliptic equation with a parameter $\lambda$ is studied. Initially, the existence of two nonnegative solutions is showed for $0 < \lambda < \hat \lambda$. The first solution has a negative energy while the energy of the second one is positive for $0 < \lambda < \lambda_0$ and negative for $\lambda_0 < \lambda < \hat \lambda$. The values $\lambda_0$ and $\hat \lambda$ are given under variational form and we show that every corresponding critical point is solution of the nonlinear elliptic problem (with a suitable multiplicative term). Finally, the existence of two classes of infinitely many solutions is showed via the Lusternik-Schnirelman theory.
Effects of small viscosity and far field boundary conditions for hyperbolic systems
Huey-Er Lin , Jian-Guo Liu and  Wen-Qing Xu
2004, 3(2): 267-290 doi: 10.3934/cpaa.2004.3.267 +[Abstract](30) +[PDF](234.2KB)
In this paper we study the effects of small viscosity term and the far-field boundary conditions for systems of convection-diffusion equations in the zero viscosity limit. The far-field boundary conditions are classified and the corresponding solution structures are analyzed. It is confirmed that the Neumann type of far-field boundary condition is preferred. On the other hand, we also identify a class of improperly coupled boundary conditions which lead to catastrophic reflection waves dominating the inlet in the zero viscosity limit. The analysis is performed on the linearized convection-diffusion model which well describes the behavior at the far field for many physical and engineering systems such as fluid dynamical equations and electro-magnetic equations. The results obtained here should provide some theoretical guidance for designing effective far field boundary conditions.
On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations
Nguyen Minh Man and  Nguyen Van Minh
2004, 3(2): 291-300 doi: 10.3934/cpaa.2004.3.291 +[Abstract](47) +[PDF](199.5KB)
This paper is concerned with the existence of almost periodic solutions of neutral functional differential equations of the form $\frac{d}{dt}Dx_t = Lx_t+f(t)$, where $D,$ $L$ are bounded linear operators from $\mathcal C$ :$ = C([-r, \quad 0],\quad \mathbb C^n )$ to $\mathbb C^n$, $f$ is an almost (quasi) periodic function. We prove that if the set of imaginary solutions of the characteristic equations is bounded and the equation has a bounded, uniformly continuous solution, then it has an almost (quasi) periodic solution with the same set of Fourier exponents as $f$.
Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index
Kotaro Tsugawa
2004, 3(2): 301-318 doi: 10.3934/cpaa.2004.3.301 +[Abstract](34) +[PDF](248.2KB)
In this paper, we treat the weakly damped, forced KdV equation on $\dot{H}^s$. We are interested in the lower bound of $s$ to assure the existence of the global attractor. The KdV equation has infinite conservation laws, each of which is defined in $H^j(j\in\mathbb Z, j\ge 0)$. The existence of the global attractor is usually proved by using those conservation laws. Because the KdV equation on $\dot{H}^s$ has no conservation law for $s<0$, it seems a natural question whether we can show the existence of the global attractor for $s<0$. Moreover, because the conservation laws restrict the behavior of solutions, the time global behavior of solutions for $s<0$ may be different from that for $s\ge 0$. By using a modified energy, we prove the existence of the global attractor for $s > -3/8$, which is identical to the global attractor for $s \ge 0$.
The global solution of an initial boundary value problem for the damped Boussinesq equation
Shaoyong Lai , Yong Hong Wu and  Xu Yang
2004, 3(2): 319-328 doi: 10.3934/cpaa.2004.3.319 +[Abstract](32) +[PDF](200.7KB)
This paper deals with an initial-boundary value problem for the damped Boussinesq equation

$u_{t t} - a u_{t t x x} - 2 b u_{t x x} = - c u_{x x x x} + u_{x x} + \beta(u^2)_{x x},$

where $ t > 0,$ $a,$ $b,$ $c$ and $\beta$ are constants. For the case $a \geq 1$ and $a+ c > b^2$, corresponding to an infinite number of damped oscillations, we derived the global solution of the equation in the form of a Fourier series. The coefficients of the series are related to a small parameter present in the initial conditions and are expressed as uniformly convergent series of the parameter. Also we prove that the long time asymptotics of the solution in question decays exponentially in time.

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