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Discrete & Continuous Dynamical Systems - A

2009 , Volume 24 , Issue 4

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Dissipative solutions for the Camassa-Holm equation
Helge Holden and Xavier Raynaud
2009, 24(4): 1047-1112 doi: 10.3934/dcds.2009.24.1047 +[Abstract](127) +[PDF](759.8KB)
We show that the Camassa--Holm equation $u_t-$uxxt+3uux-$2u_xuxx-uuxxx=0 possesses a global continuous semigroup of weak dissipative solutions for initial data $u|_{t=0}$ in $H^1$. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. Stability in terms of $H^1$ and $L^\infty$ norm is discussed.
On scattering for NLS: From Euclidean to hyperbolic space
Valeria Banica, Rémi Carles and Thomas Duyckaerts
2009, 24(4): 1113-1127 doi: 10.3934/dcds.2009.24.1113 +[Abstract](75) +[PDF](231.8KB)
We prove asymptotic completeness in the energy space for the nonlinear Schrödinger equation posed on hyperbolic space $\mathbb H^n$ in the radial case, for $n\ge 4$, and any energy-subcritical, defocusing, power nonlinearity. The proof is based on simple Morawetz estimates and weighted Strichartz estimates. We investigate the same question on spaces which sort of interpolate between Euclidean space and hyperbolic space, showing that the family of short range nonlinearities becomes larger and larger as the space approaches the hyperbolic space. Finally, we describe the large time behavior of radial solutions to the free dynamics.
On the multifractal formalism for Bernoulli products of invertible matrices
Imen Bhouri and Houssem Tlili
2009, 24(4): 1129-1145 doi: 10.3934/dcds.2009.24.1129 +[Abstract](88) +[PDF](257.4KB)
We study the multifractal formalism for Bernoulli products of invertible matrices. Using the Fourier-Laplace transform, we prove the existence of a Frostman measure and so the validity of multifractal formalism. As an application we give an estimation of the spectrum of singularities of a harmonic function defined on the Sierpiński gasket.
Damped wave equations with fast growing dissipative nonlinearities
Alexandre Nolasco de Carvalho, Jan W. Cholewa and Tomasz Dlotko
2009, 24(4): 1147-1165 doi: 10.3934/dcds.2009.24.1147 +[Abstract](130) +[PDF](292.6KB)
Let $a>0$, $\Omega\subset \R^N$ be a bounded smooth domain and $-A$ denotes the Laplace operator with Dirichlet boundary condition in $L^2(\Omega)$. We study the damped wave problem

utt$ + a u_t + A u = f(u), \ t>0, $
$u(0)=u_0\in H^1_0(\Omega), \ \ u_t(0)=v_0\in L^2(\Omega),$

where $f:\R\to\R$ is a continuously differentiable function satisfying the growth condition $|f(s)-f(t)|\leq C|s-t|(1+|s|^{\rho-1}+|t|^{\rho-1})$, $1<\rho<\frac{N+2}{N-2}$, ($N\geq 3$), and the dissipativeness condition $\lim$sup$_|s|\to\infty \frac{f(s)}{s}< \lambda_1$ with $\lambda_1$ being the first eigenvalue of $A$. We construct the global weak solutions of this problem as the limits as $\eta\to0^+$ of the solutions of wave equations involving the strong damping term $2\eta A^{1/2} u$ with $\eta>0$. We define a subclass $\mathcal LS\subset C([0,\infty),L^2(\Omega)\times H^{-1}(\Omega))\cap L^\infty([0,\infty),H^1_0(\Omega)\times L^2(\Omega))$ of the 'limit' solutions such that through each initial condition from $H^1_0(\Omega)\times L^2(\Omega)$ passes at least one solution of the class $\mathcal LS$. We show that the class $\mathcal LS$ has bounded dissipativeness property in $H^1_0(\Omega)\times L^2(\Omega)$ and we construct a closed bounded invariant subset A of $H^1_0(\Omega)\times L^2(\Omega)$, which is weakly compact in $H^1_0(\Omega)\times L^2(\Omega)$ and compact in $H^s_{\I}(\Omega)\times H^{s-1}(\Omega)$, $s\in[0,1)$. Furthermore A attracts bounded subsets of $H^1_0(\Omega)\times L^2(\Omega)$ in $H^s_\{I\}(\Omega)\times H^{s-1}(\Omega)$, for each $s\in[0,1)$. For $N=3,4,5$ we also prove a local uniqueness result for the case of smooth initial data.

An integral system and the Lane-Emden conjecture
Wenxiong Chen and Congming Li
2009, 24(4): 1167-1184 doi: 10.3934/dcds.2009.24.1167 +[Abstract](140) +[PDF](243.9KB)
We consider the system of integral equations in $R^n$:

$ u(x) = \int_{R^{n}} \frac{1}{|x - y|^{n-\mu}} v^q (y) dy$
$ v(x) = \int_{R^{n}} \frac{1}{|x - y|^{n-\mu}} u^p(y) dy$

with $0 < \mu < n$. Under some integrability conditions, we obtain radial symmetry of positive solutions by using the method of moving planes in integral forms. In the special case when $\mu = 2$, we show that the integral system is equivalent to the elliptic PDE system

-Δ $u = v^q (x)$
-Δ $v = u^p (x)$

in $R^n$. Our symmetry result, together with non-existence of radial solutions by Mitidieri [30], implies that, under our integrability conditions, the PDE system possesses no positive solution in the subcritical case. This partially solved the well-known Lane-Emden conjecture.

Thermodynamic invariants of Anosov flows and rigidity
Yong Fang
2009, 24(4): 1185-1204 doi: 10.3934/dcds.2009.24.1185 +[Abstract](93) +[PDF](250.9KB)
By using a formula relating topological entropy and cohomological pressure, we obtain several rigidity results about contact Anosov flows. For example, we prove the following result: Let $\varphi$ be a $C^\infty$ contact Anosov flow. If its Anosov splitting is $C^2$ and it is $C^0$ orbit equivalent to the geodesic flow of a closed negatively curved Riemannian manifold, then the cohomological pressure and the metric entropy of $\varphi$ coincide. This result generalizes a result of U. Hamenstädt for geodesic flows.
An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions
Sébastien Gouëzel
2009, 24(4): 1205-1208 doi: 10.3934/dcds.2009.24.1205 +[Abstract](76) +[PDF](93.2KB)
We construct a uniformly expanding map of the interval, preserving Lebesgue measure, such that the corresponding transfer operator admits a spectral gap on the space of Lipschitz functions, but does not act continuously on the space of bounded variation functions.
On a question of Katok in one-dimensional case
Yunping Jiang
2009, 24(4): 1209-1213 doi: 10.3934/dcds.2009.24.1209 +[Abstract](88) +[PDF](103.7KB)
We prove that a $C^{1}$ orientation-preserving circle endomorphism which is Hölder conjugate to a $C^{1}$ circle expanding endomorphism is itself expanding.
On the global attractor of delay differential equations with unimodal feedback
Eduardo Liz and Gergely Röst
2009, 24(4): 1215-1224 doi: 10.3934/dcds.2009.24.1215 +[Abstract](106) +[PDF](306.2KB)
We give bounds for the global attractor of the delay differential equation $ \dot x(t)=-\mu x(t)+f(x(t-\tau))$, where $f$ is unimodal and has negative Schwarzian derivative. If $f$ and $\mu$ satisfy certain condition, then, regardless of the delay, all solutions enter the domain where $f$ is monotone decreasing and the powerful results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions. In this situation we determine the sharpest interval that contains the global attractor for any delay. In the absence of that condition, improving earlier results, we show that if the delay is sufficiently small, then all solutions enter the domain where $f'$ is negative. Our theorems then are illustrated by numerical examples using Nicholson's blowflies equation and the Mackey-Glass equation.
On the meromorphic non-integrability of some $N$-body problems
Juan J. Morales-Ruiz and Sergi Simon
2009, 24(4): 1225-1273 doi: 10.3934/dcds.2009.24.1225 +[Abstract](59) +[PDF](611.5KB)
We present a proof of the meromorphic non--integrability of the planar $N$-Body Problem for some special cases. A simpler proof is added to those already existing for the Three-Body Problem with arbitrary masses. The $N$-Body Problem with equal masses is also proven non-integrable. Furthermore, a new general result on additional integrals is obtained which, applied to these specific cases, proves the non-existence of an additional integral for the general Three-Body Problem, and provides for an upper bound on the amount of additional integrals for the equal-mass Problem for $N=4,5,6$. These results appear to qualify differential Galois theory, and especially a new incipient theory stemming from it, as an amenable setting for the detection of obstructions to Hamiltonian integrability.
The focusing energy-critical fourth-order Schrödinger equation with radial data
Benoît Pausader
2009, 24(4): 1275-1292 doi: 10.3934/dcds.2009.24.1275 +[Abstract](131) +[PDF](278.3KB)
In this paper, we investigate the focusing energy-critical fourth-order Schrödinger equation in the radial setting. We prove global existence and scattering for solutions of energy and $\dot{H}^2$-norm below that of the ground state.
Reflection of highly oscillatory waves with continuous oscillatory spectra for semilinear hyperbolic systems
Yahong Peng and Yaguang Wang
2009, 24(4): 1293-1306 doi: 10.3934/dcds.2009.24.1293 +[Abstract](76) +[PDF](220.2KB)
In this paper, we study an initial-boundary value problem for a semilinear hyperbolic system with the initial date having a possibly continuous oscillatory spectrum in the half-space $R^{1+2}_+=\{x=(t,y_{1},y_{2}):t>0,y_{2}>0\}. $ The goal of this paper is to rigorously justify the asymptotic analysis for the reflection of wave trains with such a continuous oscillatory spectrum.
Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on $\mathbb{R}^{3}$
Tristan Roy
2009, 24(4): 1307-1323 doi: 10.3934/dcds.2009.24.1307 +[Abstract](94) +[PDF](232.9KB)
We prove global well-posedness for the defocusing cubic wave equation

tt $u - \Delta u = -u^{3} $
$u(0,x) = u_{0}(x) $
$\partial_{t} u(0,x) = u_{1}(x)$

with data $( u_{0}, u_{1} ) \in H^{s} \times H^{s-1}$, $1 > s > \frac{13}{18} $≈ 0.722. The main task is to estimate the variation of an almost conserved quantity on an arbitrary long time interval. We divide it into subintervals. On each of these subintervals we write the solution as the sum of its linear part adapted to the subinterval and its corresponding nonlinear part. Some terms resulting from this decomposition have a controlled global variation and other terms have a slow local variation.

Robustly expansive homoclinic classes are generically hyperbolic
Martín Sambarino and José L. Vieitez
2009, 24(4): 1325-1333 doi: 10.3934/dcds.2009.24.1325 +[Abstract](69) +[PDF](161.1KB)
Let $f: M \to M$ be a diffeomorphism defined in a $d$-dimensional compact boundary-less manifold $M$. We prove that generically $C^1$-robustly expansive homoclinic classes $H(p)$, $p$ an $f$-hyperbolic periodic point, are hyperbolic.
Improved condition for stabilization of controlled inverted pendulum under stochastic perturbations
Leonid Shaikhet
2009, 24(4): 1335-1343 doi: 10.3934/dcds.2009.24.1335 +[Abstract](64) +[PDF](173.4KB)
Known sufficient condition for stabilization of the controlled inverted pendulum under stochastic perturbations is improved via V.Kolmanovskii and L.Shaikhet general method of Lyapunov functionals construction.
Stability of invariant measures
Siniša Slijepčević
2009, 24(4): 1345-1363 doi: 10.3934/dcds.2009.24.1345 +[Abstract](75) +[PDF](253.4KB)
We generalize various notions of stability of invariant sets of dynamical systems to invariant measures, by defining a topology on the set of measures. The defined topology is similar, but not topologically equivalent to weak* topology, and it also differs from topologies induced by the Riesz Representation Theorem. It turns out that the constructed topology is a solution of a limit case of a $p$-optimal transport problem, for $p=\infty$.
Semi-hyperbolic patches of solutions of the pressure gradient system
Kyungwoo Song and Yuxi Zheng
2009, 24(4): 1365-1380 doi: 10.3934/dcds.2009.24.1365 +[Abstract](100) +[PDF](271.2KB)
We construct patches of self-similar solutions, in which one family out of two nonlinear families of characteristics starts on sonic curves and ends on transonic shock waves, to the two-dimensional pressure gradient system. This type of solutions is common in the solutions of two-dimensional Riemann problems, as seen from numerical experiments. They are not determined by the hyperbolic domain of determinacy in the traditional sense. They are middle-way between the fully hyperbolic (supersonic) and elliptic region, which we call semi-hyperbolic or partially hyperbolic. Our intention is to use the patches as building tiles to construct global solutions to general Riemann problems.
Stability criteria for a class of linear differential equations with off-diagonal delays
Masakatsu Suzuki and Hideaki Matsunaga
2009, 24(4): 1381-1391 doi: 10.3934/dcds.2009.24.1381 +[Abstract](69) +[PDF](168.2KB)
This paper is concerned with a linear differential equation with off-diagonal delays. Some necessary and sufficient conditions are established for the zero solution of the equation to be asymptotically stable by means of root-analysis for its associated characteristic equation. Examples are also presented to illustrate the main result.
Dynamics of functions with an eventual negative Schwarzian derivative
Benjamin Webb
2009, 24(4): 1393-1408 doi: 10.3934/dcds.2009.24.1393 +[Abstract](59) +[PDF](761.4KB)
In the study of one dimensional dynamical systems one often assumes that the functions involved have a negative Schwarzian derivative. In this paper we consider a generalization of this condition. Specifically, we consider the interval functions of a real variable having some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class of functions. The introduction of this class was motivated by some maps arising in neuroscience.
Variational principles of pressure
Guohua Zhang
2009, 24(4): 1409-1435 doi: 10.3934/dcds.2009.24.1409 +[Abstract](140) +[PDF](296.8KB)
Given a topological dynamical system $(X, T)$, a Borel cover $\mathcal{U}$ of $X$ and a sub-additive sequence $\mathcal{F}$ of real-valued continuous functions on $X$, two notions of measure-theoretical pressure $P_\mu^- (T, \mathcal{U}, \mathcal{F})$ and $P_\mu^+ (T, \mathcal{U}, \mathcal{F})$ for an invariant Borel probability measure $\mu$ are introduced. When $\mathcal{U}$ is an open cover, a local variational principle between topological and measure-theoretical pressure is proved; it is also established the upper semi-continuity of P+$(T, \mathcal{U}, \mathcal{F})$ and P+$(T, \mathcal{U}, \mathcal{F})$ on the space of all invariant Borel probability measures. The notions of measure-theoretical pressure $P_\mu^- (T, X, \mathcal{F})$ and $P_\mu^+ (T, X, \mathcal{F})$ for an invariant Borel probability measure $\mu$ are also introduced. A global variational principle between topological and measure-theoretical pressure is also obtained.

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