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Communications on Pure & Applied Analysis

2003 , Volume 2 , Issue 2

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A Hilbert space approach to bounded analytic extension in the ball
Daniel Alpay , Mihai Putinar and  Victor Vinnikov
2003, 2(2): 139-145 doi: 10.3934/cpaa.2003.2.139 +[Abstract](30) +[PDF](169.3KB)
One proves, using methods of Hilbert spaces with a reproducing kernel, that any bounded analytic function on a complex curve in general position in the unit ball of C$^n$ extends to a function in the Schur class of the ball.
Attractiveness and Hopf bifurcation for retarded differential equations
R. Ouifki , M. L. Hbid and  O. Arino
2003, 2(2): 147-158 doi: 10.3934/cpaa.2003.2.147 +[Abstract](58) +[PDF](230.0KB)
This paper deals with attractiveness and Hopf bifurcation for functional differential equations. The method used is based on the center manifold reduction and the $h$-asymptotic stability related to the Poincaré procedure.
The multiplicity of solutions and geometry in a wave equation
Q-Heung Choi , Changbum Chun and  Tacksun Jung
2003, 2(2): 159-170 doi: 10.3934/cpaa.2003.2.159 +[Abstract](27) +[PDF](198.7KB)
We investigate multiplicity of solutions of the nonlinear one dimensional wave equation with Dirichlet boundary condition on the interval $(-\frac{\pi}{2},\frac{\pi}{2})$ and periodic condition on the variable $t.$ Our concern is to investigate a relation between multiplicity of solutions and source terms of the equation when the nonlinearity $-(bu^{+} - a u^{-})$ crosses an eigenvalue $\lambda_{10}$ and the source term $f$ is generated by three eigenfunctions.
Nonhomogeneous polyharmonic elliptic problems at critical growth with symmetric data
Mónica Clapp and  Marco Squassina
2003, 2(2): 171-186 doi: 10.3934/cpaa.2003.2.171 +[Abstract](21) +[PDF](259.3KB)
We show the existence of multiple solutions of a perturbed polyharmonic elliptic problem at critical growth with Dirichlet boundary conditions when the domain and the nonhomogenous term are invariant with respect to some group of symmetries.
Attractors in continuous –time switching networks
Tomáš Gedeon
2003, 2(2): 187-209 doi: 10.3934/cpaa.2003.2.187 +[Abstract](27) +[PDF](265.8KB)
We consider a system of equations with discontinuous right hand side, which arise as models of gene and neural networks. We study attractors in $R^4$ which lie in a set of orthants in the form of figure eight. We find that if the attractor is symmetric with respect to these two loops, then the only possible attractor is a periodic orbit which traverses both loops once. We show that without the symmetry the set of admissible attractors include periodic orbits which follow one loop $k$ times and other loop once, for any $k$. However, we also show that no trajectory in an attractor can traverse both loops more then once in a row.
Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —
Masaki Hibino
2003, 2(2): 211-231 doi: 10.3934/cpaa.2003.2.211 +[Abstract](30) +[PDF](246.4KB)
This paper is concerned with the existence of the Gevrey asymptotic solutions for the divergent formal solution of singular first order linear partial differential equations of nilpotent type. By using the Gevrey version of Borel-Ritt's theorem, we can prove the existence of asymptotic solutions in a small sector unconditionally. However, when we require the Borel summability of the formal solution (that is, the existence of asymptotic solutions in an open disk), global analytic continuation properties for coefficients are demanded.
Some results about a bidimensional version of the generalized BO
Aniura Milanés
2003, 2(2): 233-249 doi: 10.3934/cpaa.2003.2.233 +[Abstract](21) +[PDF](275.2KB)
For the bidimensional version of the generalized Benjamin-Ono equation:

$u_t-H^{(x)}u_{x y}+u^p u_y=0, \quad t\in \mathbb R,\quad (x,y)\in \mathbb R^2,$

we use the method of parabolic regularization to prove local well-posedness in the spaces $H^s(\mathbb R^2), \quad s>2$ and in the weighted spaces $\mathcal F_r^s=H^s(\mathbb R^2) \cap L^2((1+x^2+y^2)^rdxdy), \quad s>2,\quad r\in [0,1]$ and $\mathcal F_{1,k}^k=H^k(\mathbb R^2) \cap L^2((1+x^2+y^{2k})dxdy), \quad k\in\mathbb N, \quad k\geq 3. \quad $ As in the case of BO there is lack of persistence for both the linear and nonlinear equations (for $p$ odd) in $\mathcal F_2^s$. That leads to unique continuation principles in a natural way. By standard methods based on $L^p-L^q$ estimates of the associated group we obtain global well-posedness for small initial data and nonlinear scattering for $p\geq 3,\quad s>3$. Nonexistence of square integrable solitary waves of the form $u(x,y,t)=v(x,y-ct),\quad c>0, \quad p\in \{1,2\}$ is obtained using the results about existence of solitary waves of the BO and variational methods.

Examinations on a three-dimensional differentiable vector field that equals its own curl
Biao Ou
2003, 2(2): 251-257 doi: 10.3934/cpaa.2003.2.251 +[Abstract](27) +[PDF](153.3KB)
Consider a three-dimensional differentiable vector field $f$ that equals its own curl. We prove that $f$ is analytic and then establish an existence and uniqueness theorem for such a vector field satisfying a prescribed boundary condition. We also outline with a few variations Professor J. Ericksen's work on a unit vector field that equals its own curl.
Asymptotic shape of a solution for the Plasma problem in higher dimensional spaces
Masataka Shibata
2003, 2(2): 259-275 doi: 10.3934/cpaa.2003.2.259 +[Abstract](22) +[PDF](233.0KB)
We consider asymptotic shape of a solution for a semilinear elliptic equation in dimensions 3 or over, by using singular perturbation technique. The equations arise in the Plasma Problem. The solution is obtained as a global minimizer of some energy functional. Precisely energy estimates and uniqueness of a solution for limiting problem gives information about asymptotic shape of a solution.

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