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Communications on Pure & Applied Analysis

2003 , Volume 2 , Issue 3

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Low regularity stability of solitons for the KDV equation
S. Raynor and  G. Staffilani
2003, 2(3): 277-296 doi: 10.3934/cpaa.2003.2.277 +[Abstract](37) +[PDF](264.6KB)
We study the long-time stability of soliton solutions to the Korteweg-deVries equation. We consider solutions $u$ to the KdV with initial data in $H^s$, $0 \leq s < 1$, that are initially close in $H^s$ norm to a soliton. We prove that the possible orbital instability of these ground states is at most polynomial in time. This is an analogue to the $H^s$ orbital instability results of [7] for the nonlinear Schrödinger equation, and obtains the same maximal growth rate in $t$. Our argument is based on the "I-method" used in [7] and other papers of Colliander, Keel, Staffilani, Takaoka and Tao.
A nonoverlapping domain decomposition method for nonconforming finite element problems
Qingping Deng
2003, 2(3): 297-310 doi: 10.3934/cpaa.2003.2.297 +[Abstract](148) +[PDF](210.1KB)
A nonoverlapping domain decomposition method for nonconforming finite element problems of second order partial differential equations is developed and analyzed. In particular, its convergence is demonstrated and convergence rate is estimated. The method is based on a Robin boundary condition as its transmission condition together with a derivative-free transmission data updating technique on the interfaces. The method is directly presented to finite element problems without introducing any Lagrange multipliers. The method can be naturally applied to general multi-subdomain decompositions and implemented on parallel machines with local communications. The method also allows choosing subdomains very flexibly, which can be even as small as an individual element. Therefore, the method can be regarded as a bridge connecting between direct methods and iterative methods for linear systems. Finally, some numerical experiments are also presented to demonstrate the effectiveness of the method.
Entire solutions of the nonlinear eigenvalue logistic problem with sign-changing potential and absorption
Teodora-Liliana Dinu
2003, 2(3): 311-321 doi: 10.3934/cpaa.2003.2.311 +[Abstract](35) +[PDF](206.2KB)
We are concerned with positive solutions decaying to zero at infinity for the logistic equation $-\Delta u=\lambda ( V(x)u-f(u))$ in $\mathbb R^N$, where $V(x)$ is a variable potential that may change sign, $\lambda$ is a real parameter, and $f$ is an absorbtion term such that the mapping $f(t)/t$ is increasing in $(0,\infty)$. We prove that there exists a bifurcation non-negative number $\Lambda$ such that the above problem has exactly one solution if $\lambda >\Lambda$, but no such a solution exists provided $\lambda\leq\Lambda$.
Regularization of the two-body problem via smoothing the potential
G. Bellettini , G. Fusco and  G. F. Gronchi
2003, 2(3): 323-353 doi: 10.3934/cpaa.2003.2.323 +[Abstract](25) +[PDF](378.2KB)
We investigate the existence of global solutions for the two-body problem, when the particles interact with a potential of the form $\frac{1}{r^\alpha}$, for $\alpha >0$. Our solutions are pointwise limits of approximate solutions $u_\alpha(\epsilon_k,\nu_k)$ which solve the equation of motion with the regularized potential $\frac{1}{(r^2+\epsilon_k^2)^{\alpha/2}}$, and with an initial condition $\nu_k$; $(\epsilon_k,\nu_k)_k$ is a sequence converging to $(0,\overline \nu)$ as $k\to +\infty$, where $\overline \nu$ is an initial condition leading to collision in the non-regularized problem. We classify all the possible limits and we compare them with the already known solutions, in particular with those obtained in the paper [9] by McGehee using branch regularization and block regularization. It turns out that when $\alpha > 2$ the double limit exist, therefore in this case the problem can be regularized according to a suitable definition.
Cooperative controls for air traffic management
A. Marigo and  Benedetto Piccoli
2003, 2(3): 355-369 doi: 10.3934/cpaa.2003.2.355 +[Abstract](28) +[PDF](245.6KB)
We consider a model of cooperative control system, that is a system where different controls act with non conflictual purposes. For example one control is chosen in order to minimize some cost while another is designed for safety purposes. In our main application, a model of Air Traffic Management, one control minimizes the travel time while the other is applied to avoid crashes in a free flight environment.
The mathematical difficulties arise because of the use of two discontinuous feedbacks at the same time. We consider stratified feedbacks and solutions in Krasowskii sense. Two refinements of this concept are given to guarantee existence and good behavior of solutions.
Bounce on a p-Laplacian
Dimitri Mugnai
2003, 2(3): 371-379 doi: 10.3934/cpaa.2003.2.371 +[Abstract](29) +[PDF](208.0KB)
The existence of nontrivial solutions for reversed variational inequalities involving $p$-Laplace operators is proved. The solutions are obtained as limits of solutions of suitable penalizing problems.
On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows
Hideo Ikeda , Koji Kondo , Hisashi Okamoto and  Shoji Yotsutani
2003, 2(3): 381-390 doi: 10.3934/cpaa.2003.2.381 +[Abstract](34) +[PDF](179.5KB)
We consider a parameterized, nonlocally constrained boundary-value problem, whose solutions are known to yield exact solutions, called Oseen's spiral flows, of the Navier-Stokes equations. We represent all solutions explicitly in terms of elliptic functions, and clarify completely the structure of the set of all the global branches of the solutions.
On the Lyapunov functions for the solutions of the generalized Burgers equation
Ezzeddine Zahrouni
2003, 2(3): 391-410 doi: 10.3934/cpaa.2003.2.391 +[Abstract](29) +[PDF](259.7KB)
We derive new Lyapunov functions not arising from energy norm for global solutions of Generalized Burgers Equation with initial data in homogeneous Besov spaces.
Positive solutions of superlinear boundary value problems with singular indefinite weight
M. Gaudenzi , P. Habets and  F. Zanolin
2003, 2(3): 411-423 doi: 10.3934/cpaa.2003.2.411 +[Abstract](43) +[PDF](214.7KB)
In the present paper, we propose a method to deal with non-ordered lower and upper solutions in the case of ODE's with singular coefficients. As an application, we study the existence of positive solutions for a two-point boundary value problem on ]0,1[ associated to the equation $u'' + a(t) g(u) = 0,$ where the function $g: \quad \mathbb R^+\to \mathbb R^+$ is continuous with superlinear growth at infinity and the weight $a(t)$ changes sign as well as it may present some singularities at $t=0$ or $t= 1.$

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