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Discrete & Continuous Dynamical Systems - A

2012 , Volume 32 , Issue 6

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The Cauchy problem at a node with buffer
Mauro Garavello and  Paola Goatin
2012, 32(6): 1915-1938 doi: 10.3934/dcds.2012.32.1915 +[Abstract](44) +[PDF](512.5KB)
We consider the Lighthill-Whitham-Richards traffic flow model on a network composed by an arbitrary number of incoming and outgoing arcs connected together by a node with a buffer. Similar to [15], we define the solution to the Riemann problem at the node and we prove existence and well posedness of solutions to the Cauchy problem, by using the wave-front tracking technique and the generalized tangent vectors.
Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes
Boris Andreianov and  Nicolas Seguin
2012, 32(6): 1939-1964 doi: 10.3934/dcds.2012.32.1939 +[Abstract](54) +[PDF](428.5KB)
We define entropy weak solutions and establish well-posedness for the Cauchy problem for the formal equation $$\partial_t u(t,x) + \partial_x \frac{u^2}2(t,x) = - \lambda \, u(t,x)\,\delta_0(x),$$ which can be seen as two Burgers equations coupled in a non-conservative way through the interface located at $x=0$. This problem appears as an important auxiliary step in the theoretical and numerical study of the one-dimensional particle-in-fluid model developed by Lagoutière, Seguin and Takahashi [30].
    The interpretation of the non-conservative product "$ u(t,x) \, \delta_0(x)$" follows the analysis of [30]; we can describe the associated interface coupling in terms of one-sided traces on the interface. Well-posedness is established using the tools of the theory of conservation laws with discontinuous flux ([4]).
    For proving existence and for practical computation of solutions, we construct a finite volume scheme, which turns out to be a well-balanced scheme and which allows a simple and efficient treatment of the interface coupling. Numerical illustrations are given.
A profinite group invariant for hyperbolic toral automorphisms
Lennard F. Bakker and  Pedro Martins Rodrigues
2012, 32(6): 1965-1976 doi: 10.3934/dcds.2012.32.1965 +[Abstract](38) +[PDF](352.7KB)
For a hyperbolic toral automorphism, we construct a profinite completion of an isomorphic copy of the homoclinic group of its right action using isomorphic copies of the periodic data of its left action. The resulting profinite group has a natural module structure over a ring determined by the right action of the hyperbolic toral automorphism. This module is an invariant of conjugacy that provides means in which to characterize when two similar hyperbolic toral automorphisms are conjugate or not. In particular, this shows for two similar hyperbolic toral automorphisms with module isomorphic left action periodic data, that the homoclinic groups of their right actions play the key role in determining whether or not they are conjugate. This gives a complete set of dynamically significant invariants for the topological classification of hyperbolic toral automorphisms.
On the higher-dimensional multifractal analysis
Julien Barral and  Yan-Hui Qu
2012, 32(6): 1977-1995 doi: 10.3934/dcds.2012.32.1977 +[Abstract](58) +[PDF](445.2KB)
We achieve the higher-dimensional multifractal analysis for quotients of almost additive potentials on topologically mixing subshifts of finite type without restriction on the regularity of the potentials, nor on the support of the Hausdorff spectrum, for which we do not need to assume that it has a non empty interior.
On a phase field model for solid-liquid phase transitions
Sylvie Benzoni-Gavage , Laurent Chupin , Didier Jamet and  Julien Vovelle
2012, 32(6): 1997-2025 doi: 10.3934/dcds.2012.32.1997 +[Abstract](30) +[PDF](572.8KB)
A new phase field model is introduced, which can be viewed as a nontrivial generalisation of what is known as the Caginalp model. It involves in particular nonlinear diffusion terms. By formal asymptotic analysis, it is shown that in the sharp interface limit it still yields a Stefan-like model with: 1) a generalized Gibbs-Thomson relation telling how much the interface temperature differs from the equilibrium temperature when the interface is moving or/and is curved with surface tension; 2) a jump condition for the heat flux, which turns out to depend on the latent heat and on the velocity of the interface with a new, nonlinear term compared to standard models. From the PDE analysis point of view, the initial-boundary value problem is proved to be locally well-posed in time (for smooth data).
Density of orbits in laminations and the space of critical portraits
Alexander Blokh , Clinton Curry and  Lex Oversteegen
2012, 32(6): 2027-2039 doi: 10.3934/dcds.2012.32.2027 +[Abstract](24) +[PDF](354.5KB)
Thurston introduced $\sigma_d$-invariant laminations (where $\sigma_d(z)$ coincides with $z^d:\mathbb{S}\to \mathbb{S}$, $d\ge 2$). He defined wandering $k$-gons as sets $T\subset \mathbb{S}$ such that $\sigma_d^n(T)$ consists of $k\ge 3$ distinct points for all $n\ge 0$ and the convex hulls of all the sets $\sigma_d^n(T)$ in the plane are pairwise disjoint. Thurston proved that $\sigma_2$ has no wandering $k$-gons and posed the problem of their existence for $\sigma_d$, $d\ge 3$.
    Call a lamination with wandering $k$-gons a WT-lamination. Denote the set of cubic critical portraits by $\mathcal{A}_3$. A critical portrait, compatible with a WT-lamination, is called a WT-critical portrait; let $\mathcal{WT}_3$ be the set of all of them. It was recently shown by the authors that cubic WT-laminations exist and cubic WT-critical portraits, defining polynomials with condense orbits of vertices of order three in their dendritic Julia sets, are dense and locally uncountable in $\mathcal{A}_3$ ($D\subset X$ is condense in $X$ if $D$ intersects every subcontinuum of $X$). Here we show that $\mathcal{WT}_3$ is a dense first category subset of $\mathcal{A}_3$, that critical portraits, whose laminations have a condense orbit in the topological Julia set, form a residual subset of $\mathcal{A}_3$, and that the existence of a condense orbit in the Julia set $J$ implies that $J$ is locally connected.
Absolute and delay-dependent stability of equations with a distributed delay
Elena Braverman and  Sergey Zhukovskiy
2012, 32(6): 2041-2061 doi: 10.3934/dcds.2012.32.2041 +[Abstract](29) +[PDF](315.8KB)
We study delay-independent stability in nonlinear models with a distributed delay which have a positive equilibrium. Models with a unique positive equilibrium frequently occur in population dynamics and other applications. In particular, we construct a relevant difference equation such that its stability implies stability of the equation with a distributed delay and a finite memory. This result is, generally speaking, incorrect for systems with infinite memory. If the relevant difference equation is unstable, we describe the general delay-independent lower and upper solution bounds and also demonstrate that the equation with a distributed delay is stable for small enough delays.
Energy cascades for NLS on the torus
Rémi Carles and  Erwan Faou
2012, 32(6): 2063-2077 doi: 10.3934/dcds.2012.32.2063 +[Abstract](43) +[PDF](804.0KB)
We consider the nonlinear Schrödinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special small initial data containing only five modes, we exhibit a countable set of time layers in which arbitrarily large modes are created. The proof relies on a reduction to multiphase weakly nonlinear geometric optics, and on the study of a particular two-dimensional discrete dynamical system.
A minimal approach to the theory of global attractors
Vladimir V. Chepyzhov , Monica Conti and  Vittorino Pata
2012, 32(6): 2079-2088 doi: 10.3934/dcds.2012.32.2079 +[Abstract](58) +[PDF](375.7KB)
For a semigroup $S(t):X\to X$ acting on a metric space $(X,d)$, we give a notion of global attractor based only on the minimality with respect to the attraction property. Such an attractor is shown to be invariant whenever $S(t)$ is asymptotically closed. As a byproduct, we generalize earlier results on the existence of global attractors in the classical sense.
A direct proof of the Tonelli's partial regularity result
Alessandro Ferriero
2012, 32(6): 2089-2099 doi: 10.3934/dcds.2012.32.2089 +[Abstract](34) +[PDF](357.5KB)
The aim of this work is to give a simple proof of the Tonelli's partial regularity result which states that any absolutely continuous solution to the variational problem $$\min\left\{\int_a^b L(t,u(t),\dot u(t))dt: u\in{\bf W}_0^{1,1}(a,b)\right\}$$ has extended-values continuous derivative if the Lagrangian function $L(t,u,\xi)$ is strictly convex in $\xi$ and Lipschitz continuous in $u$, locally uniformly in $\xi$ (but not in $t$). Our assumption is weaker than the one used in [2, 4, 5, 6, 13] since we do not require the Lipschitz continuity of $L$ in $u$ to be locally uniform in $t$, and it is optimal as shown by the example in [12].
Quasi-periodic solutions for derivative nonlinear Schrödinger equation
Meina Gao and  Jianjun Liu
2012, 32(6): 2101-2123 doi: 10.3934/dcds.2012.32.2101 +[Abstract](67) +[PDF](469.9KB)
In this paper, we discuss the existence of time quasi-periodic solutions for the derivative nonlinear Schrödinger equation $$\label{1.1}\mathbf{i} u_t+u_{xx}+\mathbf{i} f(x,u,\bar{u})u_x+g(x,u,\bar{u})=0 $$ subject to Dirichlet boundary conditions. Using an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field and Birkhoff normal form, we will prove that there exist a Cantorian branch of KAM tori and thus many time quasi-periodic solutions for the above equation.
Generalized Stokes system in Orlicz spaces
Piotr Gwiazda , Agnieszka Świerczewska-Gwiazda and  Aneta Wróblewska
2012, 32(6): 2125-2146 doi: 10.3934/dcds.2012.32.2125 +[Abstract](43) +[PDF](513.9KB)
The paper concerns the generalized Stokes system with the nonlinear term having growth conditions prescribed by an ${\mathcal{N}}-$function. Our main interest is directed to relaxing the assumptions on the ${\mathcal{N}}-$function and in particular to capture the shear thinning fluids with rheology close to linear. The case of anisotropic functions is considered. The existence of weak solutions is the main result of the present paper. Additionally, for the purpose of the existence proof, a version of the Sobolev-Korn inequality in Orlicz spaces is proved.
On the number of limit cycles in general planar piecewise linear systems
Song-Mei Huan and  Xiao-Song Yang
2012, 32(6): 2147-2164 doi: 10.3934/dcds.2012.32.2147 +[Abstract](63) +[PDF](753.5KB)
Much progress has been made in planar piecewise smooth dynamical systems. However there remain many important problems to be solved even in planar piecewise linear systems. In this paper, we investigate the number of limit cycles of planar piecewise linear systems with two linear regions sharing the same equilibrium. By studying the implicit Poincaré map induced by the discontinuity boundary, some cases when there exist at most 2 limit cycles is completely investigated. Especially, based on these results we provide an example along with numerical simulations to illustrate the existence of 3 limit cycles thus have a negative answer to the conjecture by M. Han and W. Zhang [11](J. Differ.Equations 248 (2010) 2399-2416) that piecewise linear systems with only two regions have at most 2 limit cycles.
Collasping behaviour of a singular diffusion equation
Kin Ming Hui
2012, 32(6): 2165-2185 doi: 10.3934/dcds.2012.32.2165 +[Abstract](34) +[PDF](486.6KB)
Let $0\le u_0(x)\in L^1(\mathbb{R}^2)\cap L^{\infty}(\mathbb{R}^2)$ be such that $u_0(x) =u_0(|x|)$ for all $|x|\ge r_1$ and is monotone decreasing for all $|x|\ge r_1$ for some constant $r_1>0$ and $\mbox{ess}\inf_{2{B}_{r_1}(0)}u_0\ge\mbox{ess} \sup_{R^2\setminus B_{r_2}(0)}u_0$ for some constant $r_2>r_1$. Then under some mild decay conditions at infinity on the initial value $u_0$ we will extend the result of P. Daskalopoulos, M.A. del Pino and N. Sesum [4], [6], and prove the collapsing behaviour of the maximal solution of the equation $u_t=\Delta\log u$ in $\mathbb{R}^2\times (0,T)$, $u(x,0)=u_0(x)$ in $\mathbb{R}^2$, near its extinction time $T=\int_{R^2}u_0dx/4\pi$ by a simplified method without using the Hamilton-Yau Harnack inequality.
Existence of nontrivial solutions to Polyharmonic equations with subcritical and critical exponential growth
Nguyen Lam and  Guozhen Lu
2012, 32(6): 2187-2205 doi: 10.3934/dcds.2012.32.2187 +[Abstract](46) +[PDF](455.2KB)
The main purpose of this paper is to establish the existence of nontrivial solutions to semilinear polyharmonic equations with exponential growth at the subcritical or critical level. This growth condition is motivated by the Adams inequality [1] of Moser-Trudinger type. More precisely, we consider the semilinear elliptic equation \[ \left( -\Delta\right) ^{m}u=f(x,u), \] subject to the Dirichlet boundary condition $u=\nabla u=...=\nabla^{m-1}u=0$, on the bounded domains $\Omega\subset \mathbb{R}^{2m}$ when the nonlinear term $f$ satisfies exponential growth condition. We will study the above problem both in the case when $f$ satisfies the well-known Ambrosetti-Rabinowitz condition and in the case without the Ambrosetti-Rabinowitz condition. This is one of a series of works by the authors on nonlinear equations of Laplacian in $\mathbb{R}^2$ and $N-$Laplacian in $\mathbb{R}^N$ when the nonlinear term has the exponential growth and with a possible lack of the Ambrosetti-Rabinowitz condition (see [23], [24]).
On dynamical behavior of viscous Cahn-Hilliard equation
Desheng Li and  Xuewei Ju
2012, 32(6): 2207-2221 doi: 10.3934/dcds.2012.32.2207 +[Abstract](41) +[PDF](417.4KB)
In this paper, we consider the initial and Dirichlet boundary value problem of the viscous Cahn-Hilliard equation with a general nonlinearity $f$, that is $$ d((1-\alpha)u-\alpha\Delta u)+(\Delta^2u-\Delta f(u))dt= 0, $$where $\alpha\in[0,1]$. Firstly, we establish the existence and continuity results on weak solutions and attractors to this problem. Secondly, we show the $\alpha$-uniform attractiveness of the attractors $A_\alpha$.
Self-maps on flat manifolds with infinitely many periods
Zhibin Liang and  Xuezhi Zhao
2012, 32(6): 2223-2232 doi: 10.3934/dcds.2012.32.2223 +[Abstract](36) +[PDF](334.8KB)
This paper deals with the homotopical minimal period of self-maps. We obtain some conditions for self-maps on flat manifolds to guarantee that their homotopical minimal periods are infinite sets.
On the fluid dynamical approximation to the nonlinear Klein-Gordon equation
Chi-Kun Lin and  Kung-Chien Wu
2012, 32(6): 2233-2251 doi: 10.3934/dcds.2012.32.2233 +[Abstract](56) +[PDF](417.0KB)
We study the nonrelativistic, semiclassical and nonrelativistic-semiclassical limits of the (modulated) nonlinear Klein-Gordon equations from its hydrodynamical structure via WKB analysis. The nonrelativistic-semiclassical limit is proved rigorously by modulated energy method.
Symmetrical symplectic capacity with applications
Chungen Liu and  Qi Wang
2012, 32(6): 2253-2270 doi: 10.3934/dcds.2012.32.2253 +[Abstract](40) +[PDF](446.8KB)
In this paper, we first introduce the concept of symmetrical symplectic capacity for symmetrical symplectic manifolds, and by using this symmetrical symplectic capacity theory we prove that there exists at least one symmetric closed characteristic (brake orbit and $S$-invariant brake orbit are two examples) on prescribed symmetric energy surface which has a compact neighborhood with finite symmetrical symplectic capacity.
Nonradial solutions for the Klein-Gordon-Maxwell equations
Percy D. Makita
2012, 32(6): 2271-2283 doi: 10.3934/dcds.2012.32.2271 +[Abstract](47) +[PDF](344.6KB)
We study a system of a nonlinear Klein-Gordon equation coupled with Maxwell's equations. We prove the existence of nonradial solutions which are radially symmetric when restricted to a hyperplane, and either periodic or non-periodic in the orthogonal direction to that very hyperplane.
Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients
Giorgio Metafune and  Chiara Spina
2012, 32(6): 2285-2299 doi: 10.3934/dcds.2012.32.2285 +[Abstract](34) +[PDF](383.3KB)
We prove heat kernel bounds for the operator $(1+|x|^\alpha)\Delta$ in $\mathbb{R}^N$, through Nash inequalities and weighted Hardy inequalities.
Fredholm's alternative for a class of almost periodic linear systems
Massimo Tarallo
2012, 32(6): 2301-2313 doi: 10.3934/dcds.2012.32.2301 +[Abstract](36) +[PDF](354.0KB)
A Fredholm alternative is proposed for linear almost periodic equations which satisfy the Favard separation condition. The alternative is then tested in the special case, where all the solutions of the homogeneous part of the equation are bounded.
Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian
Louis Tebou
2012, 32(6): 2315-2337 doi: 10.3934/dcds.2012.32.2315 +[Abstract](41) +[PDF](175.8KB)
We consider an Euler-Bernoulli equation in a bounded domain with a local dissipation of viscoelastic type involving the $p$-Laplacian. The dissipation is effective in a suitable nonvoid subset of the domain under consideration. This equation corresponds to the plate equation with a localized structural damping when both the parameter $p$ and the space dimension equal two. First we prove existence, uniqueness, and smoothness results. Then, using an appropriate perturbed energy coupled with multiplier technique, we provide a constructive proof for the exponential and polynomial decay estimates of the underlying energy. It seems to us that this is the first time that a dissipation involving the $p$-Laplacian is used in the framework of stabilization of second order evolution equations with locally distributed damping.
Traveling curved fronts in monotone bistable systems
Zhi-Cheng Wang
2012, 32(6): 2339-2374 doi: 10.3934/dcds.2012.32.2339 +[Abstract](38) +[PDF](635.2KB)
This paper is concerned with the existence, uniqueness and stability of traveling curved fronts for reaction-diffusion bistable systems in two-dimensional space. By establishing the comparison theorem and constructing appropriate supersolutions and subsolutions, we prove the existence of traveling curved fronts. Furthermore, we show that the curved front is globally stable. Finally, we apply the results to three important models in biology.

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