# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

November 2011 , Volume 30 , Issue 4

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2011, 30(4): 995-1035 doi: 10.3934/dcds.2011.30.995 +[Abstract](752) +[PDF](654.8KB)
Abstract:
We develop the techniques of [25] and [11] in order to derive dispersive estimates for a matrix Hamiltonian equation defined by linearizing about a minimal mass soliton solution of a saturated, focussing nonlinear Schrödinger equation

$\i u_t + \Delta u + \beta (|u|^2) u = 0$
$\u(0,x) = u_0 (x),$

in $\mathbb{R}^3$. These results have been seen before, though we present a new approach using scattering theory techniques. In further works, we will numerically and analytically study the existence of a minimal mass soliton, as well as the spectral assumptions made in the analysis presented here.
2011, 30(4): 1037-1054 doi: 10.3934/dcds.2011.30.1037 +[Abstract](879) +[PDF](404.1KB)
Abstract:
Motivated by the physical theory of Critical Dynamics the Cahn-Hilliard equation on a bounded space domain is considered and forcing terms of general type are introduced. For such a rescaled equation the limiting inter-face problem is studied and the following are derived: (i) asymptotic results indicating that the forcing terms may slow down the equilibrium locally or globally, (ii) the sharp interface limit problem in the multidimensional case demonstrating a local influence in phase transitions of terms that stem from the chemical potential, while free energy independent terms act on the rest of the domain, (iii) a limiting non-homogeneous linear diffusion equation for the one-dimensional problem in the case of deterministic forcing term that follows the white noise scaling.
2011, 30(4): 1055-1081 doi: 10.3934/dcds.2011.30.1055 +[Abstract](1031) +[PDF](522.6KB)
Abstract:
Let $\Omega$ be a smooth bounded domain. We are concerned about the following nonlinear elliptic problem:

$\Delta u + |x|^{\alpha}u^{p} = 0, \ u > 0 \quad$ in $\Omega$,
$\ u = 0 \quad$ on $\partial \Omega$,

where $\alpha > 0, p \in (1,\frac{n+2}{n-2}).$ In this paper, we show that for $n \ge 8$, a maximum point $x_{\alpha }$ of a least energy solution of above problem converges to a point $x_0 \in \partial^$*$\Omega$ satisfying $H(x_0) = \min_$$\w \in \partial ^* \Omega$$H( w )$ as $\alpha \to \infty,$ where $H$ is the mean curvature on $\partial \Omega$ and $\partial ^$*$\Omega \equiv \{ x \in \partial \Omega : |x| \ge |y|$ for any $y \in \Omega \}.$
2011, 30(4): 1083-1093 doi: 10.3934/dcds.2011.30.1083 +[Abstract](1225) +[PDF](397.1KB)
Abstract:
We consider the fully nonlinear integral systems involving Wolff potentials:

$\u(x) = W_{\beta, \gamma}(v^q)(x)$, $\x \in R^n$;
$\v(x) = W_{\beta, \gamma} (u^p)(x)$, $\x \in R^n$;

(1)

where

$\W_{\beta,\gamma} (f)(x) = \int_0^{\infty}$ $[ \frac{\int_{B_t(x)} f(y) dy}{t^{n-\beta\gamma}} ]^{\frac{1}{\gamma-1}} \frac{d t}{t}.$

After modifying and refining our techniques on the method of moving planes in integral forms, we obtain radial symmetry and monotonicity for the positive solutions to systems (1).
This system includes many known systems as special cases, in particular, when $\beta = \frac{\alpha}{2}$ and $\gamma = 2$, system (1) reduces to

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

(2)

The solutions $(u,v)$ of (2) are critical points of the functional associated with the well-known Hardy-Littlewood-Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic PDEs

$(-\Delta)^{\alpha/2} u = v^q$, in $R^n$,
$(-\Delta)^{\alpha/2} v = u^p$, in $R^n$

(3)

which comprises the well-known Lane-Emden system and Yamabe equation.
2011, 30(4): 1095-1106 doi: 10.3934/dcds.2011.30.1095 +[Abstract](628) +[PDF](219.1KB)
Abstract:
We give examples of rank one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesics. We build rank one compact surfaces such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodesic flow. Finally, we give examples of complete rank one surfaces for which the non wandering set of the geodesic flow is connected, the periodic orbits are dense in that set, yet the geodesic flow is not transitive in restriction to its non wandering set.
2011, 30(4): 1107-1138 doi: 10.3934/dcds.2011.30.1107 +[Abstract](854) +[PDF](538.3KB)
Abstract:
Our aim is to study the pointwise time-asymptotic behavior of solutions for the scalar conservation laws with relaxation in multi-dimensions. We construct the Green's function for the Cauchy problem of the relaxation system which satisfies the dissipative condition. Based on the estimate for the Green's function, we get the pointwise estimate for the solution. It is shown that the solution exhibits some weak Huygens principle where the characteristic 'cone' is the envelope of planes.
2011, 30(4): 1139-1144 doi: 10.3934/dcds.2011.30.1139 +[Abstract](892) +[PDF](312.5KB)
Abstract:
N/A
2011, 30(4): 1145-1159 doi: 10.3934/dcds.2011.30.1145 +[Abstract](674) +[PDF](435.4KB)
Abstract:
Under the Coulomb gauge condition Chern-Simons-Higgs equations are formulated in the hyperbolic system coupled with elliptic equations. We consider a solution of Chern-Simons-Higgs equations with finite energy and show how to obtain $H^1$ solution with one exceptional term $\phi\partial_t A_0$ from which the model equations (63) are proposed.
2011, 30(4): 1161-1180 doi: 10.3934/dcds.2011.30.1161 +[Abstract](724) +[PDF](462.7KB)
Abstract:
This paper is concerned with a modified two-component periodic Camassa-Holm system. The local well-posedness and low regularity result of solution are established by using the techniques of pseudoparabolic regularization and some priori estimates derived from the equation itself. A wave-breaking for strong solutions and several results of blow-up solution with certain initial profiles are described. In addition, the initial boundary value problem for a modified two-component periodic Camassa-Holm system is also considered.
2011, 30(4): 1181-1189 doi: 10.3934/dcds.2011.30.1181 +[Abstract](865) +[PDF](381.7KB)
Abstract:
For mixing $\mathbb Z^d$-actions generated by commuting automorphisms of a compact abelian group, we investigate the directional uniformity of the rate of periodic point distribution and mixing. When each of these automorphisms has finite entropy, it is shown that directional mixing and directional convergence of the uniform measure supported on periodic points to Haar measure occurs at a uniform rate independent of the direction.
2011, 30(4): 1191-1210 doi: 10.3934/dcds.2011.30.1191 +[Abstract](716) +[PDF](337.9KB)
Abstract:
We propose new Kruzhkov type entropy conditions for one dimensional scalar conservation law with a discontinuous flux. We prove existence and uniqueness of the entropy admissible weak solution to the corresponding Cauchy problem merely under assumptions on the flux which provide the maximum principle. In particular, we allow multiple flux crossings and we do not need any kind of genuine nonlinearity conditions.
2011, 30(4): 1211-1235 doi: 10.3934/dcds.2011.30.1211 +[Abstract](691) +[PDF](946.9KB)
Abstract:
Retroreflectors are optical devices that reverse the direction of incident beams of light. Here we present a collection of billiard type retroreflectors consisting of four objects; three of them are asymptotically perfect retroreflectors, and the fourth one is a retroreflector which is very close to perfect. Three objects of the collection have recently been discovered and published or submitted for publication. The fourth object --- notched angle --- is a new one; a proof of its retroreflectivity is given.
2011, 30(4): 1237-1242 doi: 10.3934/dcds.2011.30.1237 +[Abstract](799) +[PDF](303.1KB)
Abstract:
We construct a pair of equivalent flows with fixed points, such that one has infinite topological entropy and the other has zero topological entropy.
2011, 30(4): 1243-1248 doi: 10.3934/dcds.2011.30.1243 +[Abstract](678) +[PDF](269.9KB)
Abstract:
In 2004, Manning showed that the topological entropy of the geodesic flow for a surface of negative curvature decreases as the metric evolves under the normalised Ricci flow. It is an interesting open problem, also due to Manning, to determine to what extent such behaviour persists for higher dimensional manifolds. In this short note, we describe the problem and give a curvature criterion under which monotonicity of the topological entropy can be established for a short time. In particular, the criterion applies to metrics of negative sectional curvature which are in the same conformal class as a metric of constant negative sectional curvature.
2011, 30(4): 1249-1262 doi: 10.3934/dcds.2011.30.1249 +[Abstract](843) +[PDF](415.7KB)
Abstract:
This paper is concerned with the following periodic Hamiltonian elliptic system

$\-\Delta \varphi+V(x)\varphi=G_\psi(x,\varphi,\psi)$ in $\mathbb{R}^N,$
$\-\Delta \psi+V(x)\psi=G_\varphi(x,\varphi,\psi)$ in $\mathbb{R}^N,$
$\varphi(x)\to 0$ and $\psi(x)\to0$ as $|x|\to\infty.$

Assuming the potential $V$ is periodic and $0$ lies in a gap of $\sigma(-\Delta+V)$, $G(x,\eta)$ is periodic in $x$ and superquadratic in $\eta=(\varphi,\psi)$, existence and multiplicity of solutions are obtained via variational approach.
2011, 30(4): 1263-1283 doi: 10.3934/dcds.2011.30.1263 +[Abstract](867) +[PDF](467.0KB)
Abstract:
In this paper, we study the asymptotic behavior of solutions to one-dimensional compressible Navier-Stokes equations with gravity and vacuum for isentropic flows with density-dependent viscosity $\mu(\rho)=c\rho^{\theta}$. Under some suitable assumptions on the initial date and $\gamma>1$, if $\theta\in(0,\frac{\gamma}{2}]$, we prove the weak solution $(\rho(x,t),u(x,t))$ behavior asymptotically to the stationary one by adapting and modifying the technique of weighted estimates. This result improves the one in [5] where Duan showed that the weak solution converges to the stationary one in the sense of integral for shallow water model. In addition, if $\theta\in(0,\frac{\gamma}{2}]\cap(0,\gamma-1]$, following the same idea in [9], we estimate the stabilization rate of the solution as time tends to infinity in the sense of $L^\infty$ norm, weighted $L^2$ norm and weighted $H^1$ norm.

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