
ISSN:
1078-0947
eISSN:
1553-5231
All Issues
Discrete and Continuous Dynamical Systems
November 2009 , Volume 24 , Issue 4
Select all articles
Export/Reference:
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.
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.
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.
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.
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.
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.
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.
We prove that a $C^{1}$ orientation-preserving circle endomorphism which is Hölder conjugate to a $C^{1}$ circle expanding endomorphism is itself expanding.
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.
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.
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.
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.
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.
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.
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.
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 $p=\infty$.
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.
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.
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.
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.
2020
Impact Factor: 1.392
5 Year Impact Factor: 1.610
2020 CiteScore: 2.2
Readers
Authors
Editors
Referees
Librarians
Special Issues
Email Alert
Add your name and e-mail address to receive news of forthcoming issues of this journal:
[Back to Top]