American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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Discrete & Continuous Dynamical Systems

September 2009 , Volume 25 , Issue 3

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2009, 25(3): 719-750 doi: 10.3934/dcds.2009.25.719 +[Abstract](2028) +[PDF](475.6KB)
Abstract:
We consider the static wall approximation to the dynamics of a particle bouncing on a periodically oscillating infinitely heavy plate while subject to a potential force. We assume the case of a potential given by a power of the height of the particle and sinusoidal motions of the plate. We find that for powers smaller than 1 the set of escaping orbits has full Hausdorff dimension for all motions and we obtain existence of elliptic islands of period 2 for arbitrarily high energies for a full-measure set of motions. Moreover, we find conditions on the potential to ensure that the total (Lebesgue) measure of elliptic islands of period 2 is either finite or infinite.
2009, 25(3): 751-775 doi: 10.3934/dcds.2009.25.751 +[Abstract](2851) +[PDF](310.4KB)
Abstract:
We study changes of variable, called time transformations, which reduce a delay differential equation (DDE) with a variable non-vanishing delay and an unbounded lag function to another DDE with a constant delay. By using this reduction, we can easily obtain a superconvergent integration of the original equation, even in the case of a non-strictly-increasing lag function, and study the type of decay to zero of solutions of scalar linear non-autonomous equations with a strictly increasing lag function.
2009, 25(3): 777-795 doi: 10.3934/dcds.2009.25.777 +[Abstract](2101) +[PDF](255.2KB)
Abstract:
We consider isometric dynamical systems on a Legendre set of a discrete valuation domain with finite residual field. We characterize the minimality of the system by using the structure sequence of the Legendre set and also find the corresponding adding machine to which such a minimal system is conjugate. The minimality of affine maps acting on the domain or on the group of units is fully studied.
2009, 25(3): 797-821 doi: 10.3934/dcds.2009.25.797 +[Abstract](2461) +[PDF](274.9KB)
Abstract:
We consider a coupled system of evolution equations modeling the propagation of elastic waves interacting with a magnetic field in a bounded simply connected region of $\mathbb{R}^3$ with boundary of class $C^2$. A nonlinear dissipative mechanism is allowed to be effective in an small subregion of $\Omega$. We prove that the total energy decays as $t \to +\infty$.
2009, 25(3): 823-842 doi: 10.3934/dcds.2009.25.823 +[Abstract](2833) +[PDF](249.1KB)
Abstract:
Hopf bifurcation in the presence of the symmetric group $S_n$ (acting naturally by permutation of coordinates) is a problem with relevance to coupled oscillatory systems. To study this bifurcation it is important to construct the Taylor expansion of the equivariant vector field in normal form. We derive generating functions for the numbers of linearly independent invariants and equivariants of any degree, and obtain recurrence relations for these functions. This enables us to determine the number of invariants and equivariants for all $n$, and show that this number is independent of $n$ for sufficiently large $n$. We also explicitly construct the equivariants of degree three and degree five, which are valid for arbitrary $n$.
2009, 25(3): 843-858 doi: 10.3934/dcds.2009.25.843 +[Abstract](1818) +[PDF](207.6KB)
Abstract:
A class of ultraparabolic equations arising from age-dependent population diffusion is analyzed. For such problems existence and uniqueness results as well as continuous dependence upon the data are proved. The regularity of solutions with respect to space variables is also proved, using the theory of interpolation spaces generated by analytic semigroups.
2009, 25(3): 859-867 doi: 10.3934/dcds.2009.25.859 +[Abstract](2090) +[PDF](153.5KB)
Abstract:
We consider the hydrodynamic theory of liquid crystals. We prove some regularity criteria for a simplified Ericksen-Leslie system. The existence and uniqueness of global smooth solutions is also proved for a regularization model of this simplified system.
2009, 25(3): 869-882 doi: 10.3934/dcds.2009.25.869 +[Abstract](2207) +[PDF](194.0KB)
Abstract:
We study the acoustic limit from the Boltzmann equation in the framework of classical solutions. For a solution $F_\varepsilon=\mu +\varepsilon \sqrt{\mu}f_\varepsilon$ to the rescaled Boltzmann equation in the acoustic time scaling

$\partial_t F_\varepsilon +\v$•$grad$x$F_\varepsilon =\frac{1}{\varepsilon} \Q(F_\varepsilon,F_\varepsilon)\,$

inside a periodic box $\mathbb{T}^3$, we establish the global-in-time uniform energy estimates of $f_\varepsilon$ in $\varepsilon$ and prove that $f_\varepsilon$ converges strongly to $f$ whose dynamics is governed by the acoustic system. The collision kernel $\Q$ includes hard-sphere interaction and inverse-power law with an angular cutoff.

2009, 25(3): 883-913 doi: 10.3934/dcds.2009.25.883 +[Abstract](2371) +[PDF](392.1KB)
Abstract:
An orbit behavior of a Hamiltonian system in two degrees of freedom in a neighborhood of a quadratically tangent homoclinic orbit to a saddle-focus equilibrium is studied, the orbit structure within the level of the saddle-focus is described. We show the existence of an intrinsic parameter (a modulus) which determines types of invariant nonuniform hyperbolic subsets. These subsets are described by means of symbolic dynamics with countably many states. Multi-round tangent and transversal homoclinic orbits to the saddle-focus have also shown to exist.
2009, 25(3): 915-931 doi: 10.3934/dcds.2009.25.915 +[Abstract](2133) +[PDF](301.8KB)
Abstract:
The existence of travelling heteroclinic waves for the sine-Gordon lattice is proved for a linear interaction of neighbouring atoms. The asymptotic states are chosen such that the action functional is finite. The proof relies on a suitable concentration-compactness argument, which can be shown to hold even though the associated functional has no sub-additive structure.
2009, 25(3): 933-949 doi: 10.3934/dcds.2009.25.933 +[Abstract](2092) +[PDF](238.5KB)
Abstract:
In this paper we study the initial boundary value problems for scalar conservation laws with source terms possessing limited regularity. We first define a strong trace of large class of entropy solutions of scalar conservation laws with source terms at the boundary $(0,T)\times\{0\}$ reached by $L^1$ in order to find a good boundary condition and we prove the well-posedness for scalar conservation laws with source terms. The proof is based on the kinetic formulation and the compensated compactness method.
2009, 25(3): 951-962 doi: 10.3934/dcds.2009.25.951 +[Abstract](2184) +[PDF](192.6KB)
Abstract:
We investigate the open Closing Lemma problem for vector fields on the $2$-dimensional torus. The local $C^r$ Closing Lemma is verified for smooth vector fields that are area-preserving at all saddle points. Namely, given such a $C^r$ vector field $X$, $r\geq 4$, with a non-trivially recurrent point $p$, there exists a vector field $Y$ arbitrarily near to $X$ in the $C^r$ topology and obtained from $X$ by a twist perturbation, such that $p$ is a periodic point of $Y$.
The proof relies on a new result in $1$-dimensional dynamics on the non-existence of semi-wandering intervals of smooth order-preserving circle maps.
2009, 25(3): 963-979 doi: 10.3934/dcds.2009.25.963 +[Abstract](2789) +[PDF](234.8KB)
Abstract:
Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain, and $H_0^1(\Omega)$ be the standard Sobolev space. Define for any $p>1$,

$\lambda_p(\Omega)=$i n f$_u\in H_0^1(\Omega),$u≠0(for some u)$\|\|\nabla u\|\|_2^2/\|\|u\|\|_p^2,$

where $|\|\cdot\||_p$ denotes $L^p$ norm. We derive in this paper a sharp form of the following improved Moser-Trudinger inequality involving the $L^p$-norm using the method of blow-up analysis:

$s u p_{u\in H_0^1(\Omega),\|\|\nabla u\|\|_2=1}\int_{\Omega} e^{4\pi (1+\alpha\|\|u\|\|_p^2)u^2}dx<+\infty$

for $0\leq \alpha <\lambda_p(\Omega)$, and the supremum is infinity for all $\alpha\geq \lambda_p(\Omega)$. We also prove the existence of the extremal functions for this inequality when $\alpha$ is sufficiently small.

2009, 25(3): 981-989 doi: 10.3934/dcds.2009.25.981 +[Abstract](2103) +[PDF](170.3KB)
Abstract:
This paper deals with the problem of finding the range of entropy values resulting from actions of discrete amenable groups by automorphisms of compact abelian groups. When the acting group $G$ is locally normal, we obtain an entropy formula and show that the full range of entropy values $[0,\infty]$ occurs for actions of $G$. We consider related entropy range problems, give sufficient conditions for zero entropy and, as a consequence, verify the known relationship between completely positive entropy and mixing for these actions.
2009, 25(3): 991-1001 doi: 10.3934/dcds.2009.25.991 +[Abstract](2304) +[PDF](190.2KB)
Abstract:
In this paper, we discuss the asymptotic stability of singular steady states of the nonlinear heat equation $u_t=\Delta u+u^p$ in weighted $L^r$ - norms.
2009, 25(3): 1003-1032 doi: 10.3934/dcds.2009.25.1003 +[Abstract](2017) +[PDF](338.8KB)
Abstract:
In this paper we give some sharp qualitative and quantitative conditions to guarantee that the exponential type of a linear nonautonomous equation is modified by a linear perturbation. No assumption (periodic, almost periodic, quasi periodic etc) is made on the time behavior of the coefficients of the equation or the perturbation.
The results are applied to the study of the asymptotic behavior, both forwards and pullback, of nonautonomous nonlinear equations.
2009, 25(3): 1033-1039 doi: 10.3934/dcds.2009.25.1033 +[Abstract](2748) +[PDF](119.3KB)
Abstract:
In this paper we establish a theory on index sum of isolated singular points of positive vector fields and present a formula of index sum of isolated singular points of a positive vector field in the positive othant of $R^n$.
2009, 25(3): 1041-1060 doi: 10.3934/dcds.2009.25.1041 +[Abstract](2539) +[PDF](249.7KB)
Abstract:
The paper deals with the nonlinear evolution equation

ε∂ttu + $\delta$∂tu+ $\omega$∂xxxx u-$[\beta+\int_0^1[\partial_y u(y,t)]^2\d y ]$∂xxu=f,

which describes the motion of the vertical deflection of an extensible Kirchhoff beam. The existence of the global attractor of optimal regularity is shown, as well as the existence of a family of exponential attractors Hölder-continuous in the symmetric Hausdorff distance (with respect to ε) and of finite fractal dimension uniformly bounded with respect to ε.

2009, 25(3): 1061-1079 doi: 10.3934/dcds.2009.25.1061 +[Abstract](2111) +[PDF](241.4KB)
Abstract:
The problem considered in this paper is the protection from overheating of a thermal conductor $\Omega_1$ by a thin anisotropic coating $\Omega_2$ (e.g. a space shuttle painted with a nano-insulator). We assume Newton's Cooling Law, so the temperature satisfies the Robin boundary condition on the outer boundary of the coating. Since the temperature function on $\Omega=\overline{\Omega}_1\cup\Omega_2$ can be expanded in terms of the eigenvalues and eigenfunctions of the elliptic operator $u\mapsto -\nabla (A \nabla u)$ with the Robin boundary condition on $\partial\Omega$, where $A$ is the thermal tensor of $\Omega$, we propose the following means to ensure the insulating ability of $\Omega_2$: (A) as many eigenvalues as possible should be small, in particular, the first eigenvalue should be small, (B) the first normalized eigenfunction should take large values on the body $\Omega_1$; we also argue that it is helpful for the understanding of the dynamics if (C) higher normalized eigenfunctions take small absolute values on $\Omega_1$. We assume that the thermal conductivity of $\Omega_2$ is small either in all directions or at least in the direction normal to $\partial\Omega_1$ (the case of "optimally aligned coating"). We study the asymptotic behavior of Robin eigenpairs as outcome of the interplay of the thermal tensor $A$, the thickness of $\Omega_2$ and the thermal transport coefficient in the Robin boundary condition, in the singular limit when either the thermal conductivity of $\Omega_2$, or the thickness of $\Omega_2$, or the thermal transport coefficient approaches $0$. By doing so, we identify the parameter ranges in which some or all of (A)-(C) occur.

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