ISSN:
1534-0392
eISSN:
1553-5258
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Communications on Pure & Applied Analysis
September 2003 , Volume 2 , Issue 3
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2003, 2(3): 277-296
doi: 10.3934/cpaa.2003.2.277
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Abstract:
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.
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.
2003, 2(3): 297-310
doi: 10.3934/cpaa.2003.2.297
+[Abstract](2540)
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Abstract:
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.
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.
2003, 2(3): 311-321
doi: 10.3934/cpaa.2003.2.311
+[Abstract](1917)
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Abstract:
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 $\lambda\leq\Lambda$.
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 $\lambda\leq\Lambda$.
2003, 2(3): 323-353
doi: 10.3934/cpaa.2003.2.323
+[Abstract](1645)
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Abstract:
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.
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.
2003, 2(3): 355-369
doi: 10.3934/cpaa.2003.2.355
+[Abstract](1692)
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Abstract:
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.
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.
2003, 2(3): 371-379
doi: 10.3934/cpaa.2003.2.371
+[Abstract](2028)
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Abstract:
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.
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.
2003, 2(3): 381-390
doi: 10.3934/cpaa.2003.2.381
+[Abstract](1904)
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Abstract:
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.
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.
2003, 2(3): 391-410
doi: 10.3934/cpaa.2003.2.391
+[Abstract](2126)
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Abstract:
We derive new Lyapunov functions not arising from energy norm for global solutions of Generalized Burgers Equation with initial data in homogeneous Besov spaces.
We derive new Lyapunov functions not arising from energy norm for global solutions of Generalized Burgers Equation with initial data in homogeneous Besov spaces.
2003, 2(3): 411-423
doi: 10.3934/cpaa.2003.2.411
+[Abstract](1951)
+[PDF](214.7KB)
Abstract:
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 $t= 1.$
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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