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Discrete & Continuous Dynamical Systems - A

2005 , Volume 12 , Issue 1

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Global well-posedness of the viscous Boussinesq equations
Thomas Y. Hou and Congming Li
2005, 12(1): 1-12 doi: 10.3934/dcds.2005.12.1 +[Abstract](552) +[PDF](203.5KB)
Abstract:
We prove the global well-posedness of the viscous incompressible Boussinesq equations in two spatial dimensions for general initial data in $H^m$ with $m\ge 3$. It is known that when both the velocity and the density equations have finite positive viscosity, the Boussinesq system does not develop finite time singularities. We consider here the challenging case when viscosity enters only in the velocity equation, but there is no viscosity in the density equation. Using sharp and delicate energy estimates, we prove global existence and strong regularity of this viscous Boussinesq system for general initial data in $H^m$ with $m \ge 3$.
On uniqueness of positive entire solutions and other properties of linear parabolic equations
Peter Poláčik
2005, 12(1): 13-26 doi: 10.3934/dcds.2005.12.13 +[Abstract](375) +[PDF](216.9KB)
Abstract:
We give a simple proof of the uniqueness, up to scalar multiples, of positive entire solutions of linear nonautonomous parabolic equations. The proof is based on a new result on exponential growth of certain expressions involving solutions of the adjoint equation. We also discuss the relation of this result to the exponential separation theorem.
On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging
V. V. Chepyzhov, M. I. Vishik and W. L. Wendland
2005, 12(1): 27-38 doi: 10.3934/dcds.2005.12.27 +[Abstract](379) +[PDF](162.3KB)
Abstract:
We study the global attractors for the dissipative sine--Gordon type wave equation with time dependent external force $g(x,t)$. We assume that the function $g(x,t)$ is translationary compact in $L^{l o c}_2(\mathbb R,L_2 (\Omega))$ and the nonlinear function $f(u)$ is bounded and satisfies a global Lipschitz condition. If the Lipschitz constant $K$ is smaller than the first eigenvalue of the Laplacian with homogeneous Dirichlet conditions and the dissipation coefficient is large, then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the wave equation. Moreover, the attractor attracts all the solutions of the equation with exponential rate.
We also consider the wave equation with rapidly oscillating external force $g^\varepsilon(x,t)=g(x,t,t/\varepsilon)$ having the average $g^0(x,t)$ as $\varepsilon\to 0+$. We assume that the function $g(x,t,\zeta)-g^0(x,t)$ has a bounded primitive with respect to $\zeta$. Then we prove that the Hausdorff distance between the global attractor $\mathcal A_\varepsilon$ of the original equation and the global attractor $\mathcal A_0$ of the averaged equation is less than $O(\varepsilon^{1/2})$.
Two species competition with an inhibitor involved
Georg Hetzer and Wenxian Shen
2005, 12(1): 39-57 doi: 10.3934/dcds.2005.12.39 +[Abstract](362) +[PDF](250.0KB)
Abstract:
The dynamics of the solution flow of a two-species Lotka-Volterra competition model with an extra equation for simple inhibitor dynamics is investigated. The model fits into the abstract framework of two-species competition systems (or $K$-monotone systems), but the equilibrium representing the extinction of both species is not a repeller. This feature distinguishes our problem from the case of classical two-species competition without inhibitor (classical case for short), where a basic assumption requires that equilibrium to be a repeller. Nevertheless, several results similar to those in the classical case, such as competitive exclusion and the existence of a "thin" separatrix, are obtained, but differently from the classical case, coexistence of the two species or extinction of one of them may depend on the initial conditions. As in almost all two species competition models, the strong monotonicity of the flow (with respect to a certain order on $\mathbb R^3$) is a key ingredient for establishing the main results of the paper.
Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems
Tatsien Li and Libin Wang
2005, 12(1): 59-78 doi: 10.3934/dcds.2005.12.59 +[Abstract](547) +[PDF](245.0KB)
Abstract:
In this paper, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in a half-unbounded domain {$(t,x)|\ t\geq 0,x\geq 0$}. Under the assumption that the positive eigenvalues are weakly linearly degenerate, we obtain the global existence and uniqueness of $C^1$ solution with small and decaying initial data. Some applications are given for the system of the planar motion of an elastic string.
Statistical properties of compact group extensions of hyperbolic flows and their time one maps
Michael Field, Ian Melbourne, Matthew Nicol and Andrei Török
2005, 12(1): 79-96 doi: 10.3934/dcds.2005.12.79 +[Abstract](350) +[PDF](269.1KB)
Abstract:
Recent work of Dolgopyat shows that "typical" hyperbolic flows exhibit rapid decay of correlations. Melbourne and Török used this result to derive statistical limit laws such as the central limit theorem and the almost sure invariance principle for the time-one map of such flows.
In this paper, we extend these results to equivariant observations on compact group extensions of hyperbolic flows and their time one maps.
Necessary and sufficient conditions for existence of solutions of a variational problem involving the curl
Ana Cristina Barroso and José Matias
2005, 12(1): 97-114 doi: 10.3934/dcds.2005.12.97 +[Abstract](347) +[PDF](255.5KB)
Abstract:
We look for necessary and sufficient conditions for the existence of solutions to the minimisation problem

$ (P) \qquad\qquad\qquad $ inf $\int_\Omega f $ (curl $u(x)) dx : u \in u_{\xi_0} + W^{1,\infty}_0(\Omega;\mathbb R^3)$

where the boundary data $u_{\xi_0}$ satisfies curl$u_{\xi_0}(x)= \xi_{0}$, for $\xi_0$ a given vector in $\mathbb R^3$.

A one-parameter family of analytic Markov maps with an intermittency transition
Manuela Giampieri and Stefano Isola
2005, 12(1): 115-136 doi: 10.3934/dcds.2005.12.115 +[Abstract](343) +[PDF](289.3KB)
Abstract:
In this paper we introduce and study a one-parameter family of piecewise analytic interval maps having the tent map and the Farey map as extrema. Among other things, we construct a Hilbert space of analytic functions left invariant by the Perron-Frobenius operator of all these maps and study the transition between discrete and continuous spectrum when approaching the intermittent situation.
Quasiperiodic solutions of semilinear Liénard equations
Bin Liu
2005, 12(1): 137-160 doi: 10.3934/dcds.2005.12.137 +[Abstract](328) +[PDF](254.8KB)
Abstract:
We deal with the existence of quasi-periodic solutions in classical sense and in the generalized sense, i.e., the existence of invariant tori and Aubry-Mather sets for some semilinear differential equations

$ x'' + F_x(x,t)x'+ a^2x + \phi(x) + e(x,t) = 0, $

where $F$ and $e$ are smooth and $2\pi$-periodic in $t$ and $a>0$ is a constant. As a consequence, we also get the boundedness of all the solutions.

On generalized Benjamin type equations
Felipe Linares and M. Scialom
2005, 12(1): 161-174 doi: 10.3934/dcds.2005.12.161 +[Abstract](317) +[PDF](237.7KB)
Abstract:
We establish local and global well-posedness for the initial value problem associated to the generalized Benjamin equation and generalizations of this in the energy space. We also studied the limit process of solutions when the surface tension is becoming small. To establish these results we make use of sharp theory developed to the study of the generalized Korteweg-de Vries equation.
Local rates of Poincaré recurrence for rotations and weak mixing
Jean René Chazottes and F. Durand
2005, 12(1): 175-183 doi: 10.3934/dcds.2005.12.175 +[Abstract](347) +[PDF](214.5KB)
Abstract:
We study the lower and upper local rates of Poincaré recurrence of rotations on the circle by means of symbolic dynamics. As a consequence, we show that if the lower rate of Poincaré recurrence of an ergodic dynamical system $(X,\mathcal F, \mu, T)$ is greater or equal to 1 $\mu$-almost everywhere, then it is weakly mixing.

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