American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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Discrete & Continuous Dynamical Systems - A

April 2005 , Volume 13 , Issue 3

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2005, 13(3): 541-552 doi: 10.3934/dcds.2005.13.541 +[Abstract](730) +[PDF](259.2KB)
Abstract:
We provide quasi-periodic solutions with two frequencies $\omega\in \mathbb R^2$ for a class of completely resonant non-linear wave equations in one and two spatial dimensions and with periodic boundary conditions. This is the first existence result for quasi-periodic solutions in the completely resonant case. The main idea is to work in an appropriate invariant subspace, in order to simplify the bifurcation equation. The frequencies, close to that of the linear system, belong to an uncountable Cantor set of measure zero where no small divisor problem arises.
2005, 13(3): 553-560 doi: 10.3934/dcds.2005.13.553 +[Abstract](709) +[PDF](210.1KB)
Abstract:
For the dissipative equations of the form

$u_{t}-u_{x x}+f(x,u,u_x)=0$

we prove that the global attractor can be parametrized by a finite number of Fourier modes and that the number of modes is algebraic in parameters. This improves our earlier result [15], where the number of required modes is exponential. The method extends to equations of order higher than two.

2005, 13(3): 561-581 doi: 10.3934/dcds.2005.13.561 +[Abstract](584) +[PDF](614.5KB)
Abstract:
We study the singular limit of competition-diffusion systems in population dynamics when the initial distribution of the solution is not entirely in a domain of attraction for the system. We prove comparison principles in the viscosity sense for the solution and supersolutions to the system. By using travelling wave solutions and the distance function to interfaces, we construct a viscosity supersolution. Finally we study the dynamics of interfaces and the long time behavior of the solution for the systems with large reaction rates.
2005, 13(3): 583-622 doi: 10.3934/dcds.2005.13.583 +[Abstract](884) +[PDF](445.8KB)
Abstract:
We prove the asymptotic stability in $H^1(\mathbb R)$ of the family of solitary waves for the Benjamin-Bona-Mahony equation,

$(1-\partial^2_x)u_t+(u+u^2)_x=0.$

We prove that a solution initially close to a solitary wave, once conveniently translated, converges weakly in $H^1(\mathbb R)$, as time goes to infinity, to a possibly different solitary wave. The proof is based on a Liouville type theorem for the flow close to the solitary waves, and makes an extensive use of a monotonicity property.

2005, 13(3): 623-635 doi: 10.3934/dcds.2005.13.623 +[Abstract](658) +[PDF](260.8KB)
Abstract:
This paper concerns the Sacker-Sell spectral decomposition of a one-parametric perturbation of a non-autonomous linear Hamiltonian system with bounded solutions. Conditions ensuring the continuous variation with respect to the parameter of the spectral intervals and subbundles are established. These conditions depend on the perturbation direction and are closely related to the topological structure of the flows induced by the initial system on the real and complex Lagrange bundles.
2005, 13(3): 637-658 doi: 10.3934/dcds.2005.13.637 +[Abstract](743) +[PDF](222.2KB)
Abstract:
In this paper we develop a general approach for investigating pattern generation problems in multi-dimensional lattice models. Let $\mathcal S$ be a set of $p$ symbols or colors, $\mathbf Z_N$ a fixed finite rectangular sublattice of $\mathbf Z^d$, $d\geq 1$ and $N$ a $d$-tuple of positive integers. Functions $U:\mathbf Z^d\rightarrow \mathcal S$ and $U_N:\mathbf Z_N\rightarrow \mathcal S$ are called a global pattern and a local pattern on $\mathbf Z_N$, respectively. We introduce an ordering matrix $\mathbf X_N$ for $\Sigma_N$, the set of all local patterns on $\mathbf Z_N$. For a larger finite lattice ${\mathbf Z}_{\scriptsize\tilde{N}}$, ${\small \tilde{N}\geq N}$, we derive a recursion formula to obtain the ordering matrix ${\mathbf X}_{\scriptsize\tilde{N}}$ of $\Sigma_{\scriptsize\tilde{N}}$ from $\mathbf X_N$. For a given basic admissible local patterns set $\mathcal B\subset \Sigma_N$, the transition matrix $\mathbf T_N(\mathcal B)$ is defined. For each $\scriptsize{\tilde{N}\geq N}$, denoted by $\Sigma_{\scriptsize\tilde{N}} (\mathcal B)$ the set of all local patterns which can be generated from $\mathcal B$, the cardinal number of $\Sigma_{\scriptsize\tilde{N}} (\mathcal B)$ is the sum of entries of the transition matrix ${\mathbf T}_{\scriptsize\tilde{N}} (\mathcal B)$ which can be obtained from $\mathbf T_N(\mathcal B)$ recursively. The spatial entropy $h(\mathcal B)$ can be obtained by computing the maximum eigenvalues of a sequence of transition matrices $\mathbf T_n(\mathcal B)$. The results can be applied to study the set of global stationary solutions in various Lattice Dynamical Systems and Cellular Neural Networks.
2005, 13(3): 659-682 doi: 10.3934/dcds.2005.13.659 +[Abstract](707) +[PDF](319.8KB)
Abstract:
We show existence of a unique, regular global solution of the parabolic-elliptic system $u_t +f(t,x,u)_x+g(t,x,u)+P_x=(a(t,x) u_x)_x$ and $-P_{x x}+P=h(t,x,u,u_x)+k(t,x,u)$ with initial data $u|_{t=0} = u_0$. Here inf$_(t,x) a(t,x)>0$. Furthermore, we show that the solution is stable with respect to variation in the initial data $u_0$ and the functions $f$, $g$ etc. Explicit stability estimates are provided. The regularized generalized Camassa--Holm equation is a special case of the model we discuss.
2005, 13(3): 683-700 doi: 10.3934/dcds.2005.13.683 +[Abstract](873) +[PDF](296.9KB)
Abstract:
This paper deals with blow-up properties of the solution to a semi-linear parabolic system with nonlinear localized sources involved in a product with local terms, subject to the null Dirichlet boundary condition. We investigate the influence of localized sources and local terms on blow-up properties for this system. It will be proved that: (i) when $m, q\leq 1$ this system possesses uniform blow-up profiles. In other words, the localized terms play a leading role in the blow-up profile for this case. (ii) when $m, q>1$, this system presents single point blow-up patterns, or say that, in this time, local terms dominate localized terms in the blow-up profile. Moreover, the blow-up rate estimates in time and space are obtained, respectively.
2005, 13(3): 701-719 doi: 10.3934/dcds.2005.13.701 +[Abstract](1431) +[PDF](263.7KB)
Abstract:
The existence and structure of uniform attractors in $V$ is proved for nonautonomous 2D Navier-stokes equations on bounded domain with a new class of external forces, termed normal in $L_{l o c}^2(\mathbb R; H)$ (see Definition 3.1), which are translation bounded but not translation compact in $L_{l o c}^2(\mathbb R; H)$. To this end, some abstract results are established. First, a characterization on the existence of uniform attractor for a family of processes is presented by the concept of measure of noncompactness as well as a method to verify it. Then, the structure of the uniform attractor is obtained by constructing skew product flow on the extended phase space with weak topology. Finally, the uniform attractor of a process is identified with that of a family of processes with symbols in the closure of the translation family of the original symbol in a Banach space with weak topology.
2005, 13(3): 721-742 doi: 10.3934/dcds.2005.13.721 +[Abstract](813) +[PDF](414.9KB)
Abstract:
We deal with linear parabolic (in the sense of Petrovskii) systems of order $2b$ with discontinuous principal coefficients. A priori estimates in Sobolev and Sobolev--Morrey spaces are proved for the strong solutions by means of potential analysis and boundedness of certain singular integral operators with kernels of mixed homogeneity. As a byproduct, precise characterization of the Morrey, $BMO$ and Hölder regularity is given for the solutions and their derivatives up to order $2b-1.$
2005, 13(3): 743-753 doi: 10.3934/dcds.2005.13.743 +[Abstract](679) +[PDF](254.9KB)
Abstract:
Using linking arguments and a cohomological index theory we obtain nontrivial solutions of $p$-Laplacian problems with nonlinearities that interact with the spectrum.
2005, 13(3): 755-778 doi: 10.3934/dcds.2005.13.755 +[Abstract](693) +[PDF](323.0KB)
Abstract:
In this article we consider the Euler equations of an ideal incompressible fluid in a $2D$ and $3D$ channel and we prove the existence and uniqueness of classical solutions for all time for the $2D$ case and the local in time existence for the $3D$ case. For the $2D$ case, the proof makes use of the Schauder fixed point, and specific properties of the Green function in a channel are derived. For the $3D$ case, we use a priori estimates on some appropriate Sobolev spaces and the existence of solution follows by the Galerkin method.
2005, 13(3): 779-793 doi: 10.3934/dcds.2005.13.779 +[Abstract](704) +[PDF](268.5KB)
Abstract:
Let $G$ be a countable amenable group containing subgroups of arbitrarily large finite index. Given a polyhedron $P$ and a real number $\rho$ such that $0 \leq \rho \leq$dim$(P)$, we construct a closed subshift $X \subset P^G$ having mean topological dimension $\rho$. This shows in particular that mean topological dimension of compact metrisable $G$-spaces take all values in $[0,\infty]$.
2005, 13(3): 795-810 doi: 10.3934/dcds.2005.13.795 +[Abstract](711) +[PDF](265.0KB)
Abstract:
In this paper, we study the monotonicity of the period function of the quadratic system

$\dot x=- y + x y,\quad \dot y=x + 2 y^2-c x^2, \quad -\infty < c < +\infty.$

We show that this system has two isochronous centers for $c=1/2$, and its period function has only one critical point for $c\in(7/5, 2)$. For all other cases, the period function is monotone. This improves the results in [1].

2005, 13(3): 811-825 doi: 10.3934/dcds.2005.13.811 +[Abstract](742) +[PDF](268.9KB)
Abstract:
We consider the system obtained by D. J. Benney-G. J. Roskes and V. E. Zakharov-A. M. Rubenchik to model the interaction of low amplitude high frequency waves with low frequency, acoustic type waves. We reduce it to a nonlinear Schrödinger equation with nonlinear terms involving nonlocal terms and derivatives of the unknown. Using various smoothing effects associated to the Schrödinger group and the structure of the nonlinearity we prove that the Cauchy problem is locally well-posed in $H^s(\mathbb R^n), s>n/2$ where $n=2,3$.
2005, 13(3): 827-841 doi: 10.3934/dcds.2005.13.827 +[Abstract](661) +[PDF](243.7KB)
Abstract:
The standard upper and lower semicontinuity results for discretized attractors [22], [13], [5] are generalized for discretizations with variable stepsize. Several examples demonstrate that the limiting behaviour depends crucially on the stepsize sequence. For stepsize sequences suitably chosen, convergence to the exact attractor in the Hausdorff metric is proven. Connections to pullback attractors in cocycle dynamics are pointed out.

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