# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

2007 , Volume 17 , Issue 4

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2007, 17(4): 691-711 doi: 10.3934/dcds.2007.17.691 +[Abstract](94) +[PDF](273.2KB)
Abstract:
We study the existence and scattering of global small amplitude solutions to generalized Boussinesq (Bq) and improved modified Boussinesq (imBq) equations with nonlinear term $f(u)$ behaving as a power $u^p$ as $u \to 0$ in $\mathbb{R}^n, n \ge 1$.
2007, 17(4): 713-729 doi: 10.3934/dcds.2007.17.713 +[Abstract](90) +[PDF](250.7KB)
Abstract:
In this paper we study a metric Hopf-Lax formula looking in particular at the Carnot-Carathéodory case. We generalize many properties of the classical euclidean Hopf-Lax formula and we use it in order to get existence results for Hamilton-Jacobi-Cauchy problems satisfying a suitable Hörmander condition.
2007, 17(4): 731-738 doi: 10.3934/dcds.2007.17.731 +[Abstract](74) +[PDF](151.3KB)
Abstract:
We prove weak ergodicity theorems for non - homogeneous Markov chains $\{X_\nu\}_{\nu\geq 0}$ taking values in a finite state space $S=\{1,\cdots,n\}$ for which the family of transition matrices $\{g(x)\}_{x\in X}$ is generated from some underlying topological or measurable dynamical system $f:X\to X$. Using the projective metric of Hilbert on $\mathcal{S}=\{(x_1,\cdots,x_n)\in\mathbb R ^n : x_i\geq 0, x_1+\cdots+x_n=1\}$, the space of distributions, we form the skew-product $T:X\times\mathcal{S}\to X\times\mathcal{S}$ defined by $T(x,p)=(f(x),g(x)p)$ and show that, for continuous $g$ positive on some set, weak ergodicity for such processes is a result of the existence of a map $\gamma:X\to\mathcal{S}$ whose graph is attracting and invariant under $T$. Some results on random compositions of non-expansive maps are obtained on the way.
2007, 17(4): 739-750 doi: 10.3934/dcds.2007.17.739 +[Abstract](92) +[PDF](199.3KB)
Abstract:
The paper deals with the initial-boundary value problem for

$u_t=a(x) (\Delta u+\lambda_1 u) \qquad$ (*)

with zero Dirichlet data in a smoothly bounded domain $\Omega \subset \R^n$, $n\ge 1$. Here $a$ is positive in $\Omega$ and Hölder continuous in $\bar\Omega$, and $\lambda_1>0$ denotes the principal eigenvalue of $-\Delta$ in $\Omega$ with Dirichlet data. It is shown that if $\int_\Omega \frac{(\dist(x,\partial\Omega))^2}{a(x)}dx=\infty$ then there exist initial data in $W^{1,\infty}(\Omega)$ such that the solution of (*) is bounded but not convergent as $t\to\infty$: It has a totally ordered $\omega$-limit set which is not a singleton. Under the above condition, the occurrence of even unbounded ordered $\omega$-limit sets is demonstrated. Conversely, if $\frac{(\dist(x,\partial\Omega))^2}{a(x)}$ is integrable then any solution emanating from initial data in $W^{1,\infty}(\Omega)$ converges to some stationary solution of (*) as time approaches infinity.

2007, 17(4): 751-770 doi: 10.3934/dcds.2007.17.751 +[Abstract](79) +[PDF](263.2KB)
Abstract:
We construct solutions of the semilinear elliptic problem

$\Delta u+ |u|^{p-1}u+$ε1/2 f = 0 in Ω
u=ε1/2 g on $\partial$Ω

in a bounded smooth domain $\Omega \subset \R^N$ $(N\geq 3)$, when the exponent $p$ is supercritical and close enough to $\frac{N+2}{N-2}$. As $p\rightarrow \frac{N+2}{N-2}$, the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. As applications, we will give some existence results, in particular, when $\O$ are symmetric domains perforated with the small hole and when $f=0$ and $g=0$.

2007, 17(4): 771-786 doi: 10.3934/dcds.2007.17.771 +[Abstract](73) +[PDF](224.1KB)
Abstract:
The work treats smoothing and dispersive properties of solutions to the Schrödinger equation with magnetic potential. Under suitable smallness assumption on the potential involving scale invariant norms we prove smoothing - Strichartz estimate for the corresponding Cauchy problem. An application that guarantees absence of pure point spectrum of the corresponding perturbed Laplace operator is discussed too.
2007, 17(4): 787-806 doi: 10.3934/dcds.2007.17.787 +[Abstract](68) +[PDF](259.7KB)
Abstract:
We consider two-dimensional slow-fast systems with a layer equation exhibiting canard cycles. The canard cycles under consideration contain both a turning point and a fast orbit connecting two jump points. At both the turning point and the connecting fast orbit we suppose the presence of a parameter permitting generic breaking. Such canard cycles depend on two parameters, that we call phase parameters. We study the relaxation oscillations near the canard cycles by means of a map from the plane of phase parameters to the plane of breaking parameters.
2007, 17(4): 807-819 doi: 10.3934/dcds.2007.17.807 +[Abstract](95) +[PDF](197.5KB)
Abstract:
Exploiting the metric approach to Hamilton-Jacobi equation recently introduced by Fathi and Siconolfi [13], we prove a singular perturbation result for a general class of Hamilton-Jacobi equations. Considered in the framework of small random perturbations of dynamical systems, it extends a result due to Kamin [19] to the case of a dynamical system having several attracting points inside the domain.
2007, 17(4): 821-833 doi: 10.3934/dcds.2007.17.821 +[Abstract](89) +[PDF](187.4KB)
Abstract:
The notion of distributional chaos was introduced by Schweizer and Smítal in [Trans. Amer. Math. Soc., 344 (1994) 737] for continuous maps of a compact interval. Further, this notion was generalized to three versions $d_1C$--$d_3C$ for maps acting on general compact metric spaces (see e.g. [Chaos Solitons Fractals, 23 (2005) 1581]). The main result of [ J. Math. Anal. Appl. , 241 (2000) 181] says that a weakened version of the specification property implies existence of a two points scrambled set which exhibits a $d_1 C$ version of distributional chaos. In this article we show that much more complicated behavior is present in that case. Strictly speaking, there exists an uncountable and dense scrambled set consisting of recurrent but not almost periodic points which exhibit uniform $d_1 C$ versions of distributional chaos.
2007, 17(4): 835-865 doi: 10.3934/dcds.2007.17.835 +[Abstract](72) +[PDF](374.0KB)
Abstract:
We use the parameterization method to prove the existence and properties of one-dimensional submanifolds of the center manifold associated to the fixed point of $C^r$ maps with linear part equal to the identity. We also provide some numerical experiments to test the method in these cases.
2007, 17(4): 867-890 doi: 10.3934/dcds.2007.17.867 +[Abstract](103) +[PDF](265.1KB)
Abstract:
In this paper we study the model that the usual Maxwell's equations are supplemented with a constitution relation in which the electric displacement equals a constant time the electric field plus an internal polarization variable and the magnetic displacement equals a constant time the magnetic field plus the microscopic magnetization. Using the Galerkin method and viscosity vanishing approach, we obtain the existence of the global weak solution for the Landau-Lifshitz-Maxwell equations. The main difficulties in this study are due to the loss of compactness in the system.
2007, 17(4): 891-900 doi: 10.3934/dcds.2007.17.891 +[Abstract](84) +[PDF](156.1KB)
Abstract:
The dynamical Borel-Cantelli lemma for some interval maps is considered. For expanding maps whose derivative has bounded variation, any sequence of intervals satisfies the dynamical Borel-Cantelli lemma. If a map has an indifferent fixed point, then the dynamical Borel-Cantelli lemma does not hold even in the case that the map has a finite absolutely continuous invariant measure and summable decay of correlations.
2007, 17(4): 901-917 doi: 10.3934/dcds.2007.17.901 +[Abstract](94) +[PDF](258.8KB)
Abstract:
We use character sums to confirm several recent conjectures of V. I. Arnold on the uniformity of distribution properties of a certain dynamical system in a finite field. On the other hand, we show that some conjectures are wrong. We also analyze several other conjectures of V. I. Arnold related to the orbit length of similar dynamical systems in residue rings and outline possible ways to prove them. We also show that some of them require further tuning.

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