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## Discrete & Continuous Dynamical Systems - A

2014 , Volume 34 , Issue 11

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2014, 34(11): 4371-4388
doi: 10.3934/dcds.2014.34.4371

*+*[Abstract](518)*+*[PDF](397.0KB)**Abstract:**

We study the problem of stabilization for the acoustic system with a spatially distributed damping. Imposing various hypotheses on the structural properties of the damping term, we identify either exponential or polynomial decay of solutions with growing time. Exponential decay rate is shown by means of a time domain approach, reducing the problem to an observability inequality to be verified for solutions of the associated conservative problem. In addition, we show a polynomial stabilization result, where the proof uses a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.

2014, 34(11): 4389-4418
doi: 10.3934/dcds.2014.34.4389

*+*[Abstract](384)*+*[PDF](4918.7KB)**Abstract:**

We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural extension of the maps associated with these algorithms; prove that these natural extensions are in fact conjugate to the first return map of the geodesic flow on a related surface; and, deduce that, up to a conjugacy, almost every real number has an infinite number of common approximants for both algorithms.

2014, 34(11): 4419-4458
doi: 10.3934/dcds.2014.34.4419

*+*[Abstract](545)*+*[PDF](598.7KB)**Abstract:**

We propose a coupled system for the interaction between Cucker-Smale flocking particles and viscous compressible fluids, and present a global existence theory and time-asymptotic behavior for the proposed model in the spatial periodic domain $\mathbb{T}^3$. Our model consists of the kinetic Cucker-Smale model for flocking particles and the isentropic compressible Navier-Stokes equations for fluids, and these two models are coupled through a drag force, which is responsible for the asymptotic alignment between particles and fluid. For the asymptotic flocking behavior, we explicitly construct a Lyapunov functional measuring the deviation from the asymptotic flocking states. For a large viscosity and small initial data, we show that the velocities of Cucker-Smale particles and fluids are asymptotically aligned to the common velocity.

2014, 34(11): 4459-4486
doi: 10.3934/dcds.2014.34.4459

*+*[Abstract](343)*+*[PDF](534.1KB)**Abstract:**

The family of symmetric one sided sub-shifts in two symbols given by a sequence $a$ is studied. We analyse some of their topological properties such as transitivity, the specification property and intrinsic ergodicity. It is shown that almost every member of this family admits only one measure of maximal entropy. It is shown that the same results hold for attractors of the family of open dynamical systems arising from the doubling map with a centred symmetric hole depending on one parameter, and for the set of points that have unique $\beta$-expansion for $\beta \in (\varphi,2)$ where $\varphi$ is the Golden Ratio.

2014, 34(11): 4487-4513
doi: 10.3934/dcds.2014.34.4487

*+*[Abstract](358)*+*[PDF](492.3KB)**Abstract:**

Let $f$ be an orientation preserving circle homeomorphism with a single break point $x_b,$ i.e. with a jump in the first derivative $f'$ at the point $x_b,$ and with irrational rotation number $\rho=\rho_{f}.$ Suppose that $f$ satisfies the Katznelson and Ornstein smoothness conditions, i.e. $f'$ is absolutely continuous on $[x_b,x_b+1]$ and $f''(x)\in \mathbb{L}^{p}([0,1), d\ell)$ for some $p>1$, where $\ell$ is Lebesque measure. We prove, that the renormalizations of $f$ are approximated by linear-fractional functions in $\mathbb{C}^{1+L^{1}}$, that means, $f$ is approximated in $C^{1}-$ norm and $f''$ is appoximated in $L^{1}-$ norm. Also it is shown, that renormalizations of circle diffeomorphisms with irrational rotation number satisfying the Katznelson and Ornstein smoothness conditions are close to linear functions in $\mathbb{C}^{1+L^{1}}$- norm.

2014, 34(11): 4515-4535
doi: 10.3934/dcds.2014.34.4515

*+*[Abstract](349)*+*[PDF](432.6KB)**Abstract:**

We consider a nonlinear fourth order in space partial differential equation arising in the context of the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids.

We use the theory of operator semigroups in order to investigate the linearization of the underlying model and see that the underlying semigroup is analytic. This leads to exponential decay results for the linear homogeneous equation.

Moreover, we prove local in time well-posedness of the model under the assumption that initial data are sufficiently small by employing a fixed point argument. Global in time well-posedness is obtained by performing energy estimates and using the classical barrier method, again for sufficiently small initial data.

Additionally, we provide results concerning exponential decay of solutions of the nonlinear equation.

2014, 34(11): 4537-4553
doi: 10.3934/dcds.2014.34.4537

*+*[Abstract](351)*+*[PDF](439.8KB)**Abstract:**

We investigate the long-time behavior of weak solutions to the thin-film type equation $$v_t =(xv - vv_{xxx})_x\ ,$$ which arises in the Hele-Shaw problem. We estimate the rate of convergence of solutions to the Smyth-Hill equilibrium solution, which has the form $\frac{1}{24}(C^2-x^2)^2_+$, in the norm $$|\!|\!| f |\!|\!|_{m,1}^2 = \int_{\mathbb{R}}(1+ |x|^{2m})|f(x)|^2 \, dx + \int_{\mathbb{R}}|f_x(x)|^2 \, dx.$$ We obtain exponential convergence in the $|\!|\!| \cdot |\!|\!|_{m,1}$ norm for all $m$ with $1\leq m< 2$, thus obtaining rates of convergence in norms measuring both smoothness and localization. The localization is the main novelty, and in fact, we show that there is a close connection between the localization bounds and the smoothness bounds: Convergence of second moments implies convergence in the $H^1$ Sobolev norm. We then use methods of optimal mass transportation to obtain the convergence of the required moments. We also use such methods to construct an appropriate class of weak solutions for which all of the estimates on which our convergence analysis depends may be rigorously derived. Though our main results on convergence can be stated without reference to optimal mass transportation, essential use of this theory is made throughout our analysis.

2014, 34(11): 4555-4563
doi: 10.3934/dcds.2014.34.4555

*+*[Abstract](345)*+*[PDF](371.7KB)**Abstract:**

In this paper we prove that there exists a positive integer $k$ with the following property: Every compact $3$-manifold with boundary carries a $C^\infty$ vector field exhibiting a $C^k$-robust attractor without dominated splitting in a robust sense.

2014, 34(11): 4565-4576
doi: 10.3934/dcds.2014.34.4565

*+*[Abstract](376)*+*[PDF](366.6KB)**Abstract:**

We consider the Cauchy problem associated to the generalized Benjamin-Bona-Mahony (BBM) equation for given data in the $L^2$-based Sobolev spaces. Depending on the order of nonlinearity and dispersion, we prove that the Cauchy problem is ill-posed for data with lower order Sobolev regularity. We also prove that, in certain range of the Sobolev regularity, even if the solution exists globally in time, it fails to be smooth.

2014, 34(11): 4577-4588
doi: 10.3934/dcds.2014.34.4577

*+*[Abstract](461)*+*[PDF](417.6KB)**Abstract:**

This paper discusses asymptotic stability properties of the neutral delay differential equation \begin{eqnarray*} y'(t) = a y (t) + b y ( t - \tau ) + c y'( t - \tau ), t > 0, \\ \end{eqnarray*} where $a,\,b,\,c$ and $\tau >0$ are real scalars. We consider the exact as well as discretized delay-dependent asymptotic stability regions for this equation and describe them in terms of explicit necessary and sufficient conditions imposed on $a,\,b,\,c$ and $\tau$. Such descriptions enable us to observe some fundamental properties of these stability regions, especially with respect to stability of corresponding numerical formulae. As a consequence of our investigations, we extend existing results on this topic.

2014, 34(11): 4589-4615
doi: 10.3934/dcds.2014.34.4589

*+*[Abstract](433)*+*[PDF](531.7KB)**Abstract:**

The present work is the first one of two papers, in which we analyse systems of higher order variational equations associated to natural Hamiltonian systems with homogeneous potential of degree $k\in\mathbb{Z}\setminus \{-1,0,1\}$. Our attempt is to give necessary conditions for complete integrability which can be deduced in a framework of differential Galois theory. We show that the higher variational equations $\mathrm{VE}_p$ of order $p\geq 2$, although complicated, have a very particular algebraic structure. More precisely, we show that if $\mathrm{VE}_1$ has virtually Abelian differential Galois group (DGG), then $\mathrm{VE}_{p}$ are solvable for an arbitrary $p>1$. We proved this inductively using what we call the

*second level integrals*. Then we formulate the necessary and sufficient conditions in terms of these second level integrals for $\mathrm{VE}_{p}$ to be virtually Abelian. We apply the above conditions to potentials of degree $k=\pm 2$ considering their $\mathrm{VE}_p$ with $p>1$ along Darboux points. For $k= 2$, $\mathrm{VE}_1$ does not give any obstruction to the integrability. We show that under certain non-resonance condition, the only degree two integrable potential is the

*multidimensional harmonic oscillator*. In contrast, for degree $k=-2$ potentials, all the $\mathrm{VE}_{p}$ along Darboux points are virtually Abelian.

2014, 34(11): 4617-4645
doi: 10.3934/dcds.2014.34.4617

*+*[Abstract](474)*+*[PDF](573.3KB)**Abstract:**

We study the blow-up problem for a superlinear heat equation \begin{equation} \label{eq:P} \tag{P} \left\{ \begin{array}{ll} \partial_t u = \epsilon \Delta u + f(u), x\in\Omega, \,\,\, t>0, \\ u(x,t)=0, x\in\partial\Omega, \,\,\, t>0, \\ u(x,0)=\varphi(x)\ge 0\, (\not\equiv 0), x\in\Omega, \end{array} \right. \end{equation} where $\partial_t=\partial/\partial t$, $\epsilon>0$ is a sufficiently small constant, $N\ge 1$, $\Omega\subset {\bf R}^N$ is a domain, $\varphi\in C^2(\Omega)\cap C(\overline{\Omega})$ is a nonnegative bounded function, and $f$ is a positive convex function in $(0,\infty)$. In [10], the author of this paper and Ishige characterized the location of the blow-up set for problem (p) with $f(u)=u^p$ ($p>1$) with the aid of the invariance of the equation under some scale transformation for the solution, which played an important role in their argument. However, due to the lack of such scale invariance for problem (p), we can not apply their argument directly to problem (p). In this paper we introduce a new transformation for the solution of problem (p), which is a generalization of the scale transformation introduced in [10], and generalize the argument of [10]. In particular, we show the relationship between the blow-up set for problem (p) and pointedness of the initial function under suitable assumptions on $f$.

2014, 34(11): 4647-4669
doi: 10.3934/dcds.2014.34.4647

*+*[Abstract](423)*+*[PDF](488.7KB)**Abstract:**

In this paper, we investigate the time decay behavior to weak solution of 2D incompressible inhomogeneous Navier-Stokes equations. Granted the decay estimates, we gain a global well-posed result of these solutions.

2014, 34(11): 4671-4688
doi: 10.3934/dcds.2014.34.4671

*+*[Abstract](359)*+*[PDF](485.3KB)**Abstract:**

In this paper we study the Lane-Emden-Fowler equation $$ (P)_ \epsilon \quad \left\{ \begin{aligned} &\Delta u+|u|^{q-2}u=0\ &\hbox{in}\ \mathcal D_ \epsilon,\\ & u=0\ &\hbox{on}\ \partial\mathcal D_ \epsilon.\\ \end{aligned}\right. $$ Here $\mathcal D_ \epsilon=\mathcal D\setminus \left\{x\in \mathcal D\ :\ \mathrm{dist}(x,\Gamma_l)\le \epsilon \right\}$, $\mathcal D$ is a smooth bounded domain in $\mathbb{R}^N$, $\Gamma_l$ is an $l-$dimensional closed manifold such that $\Gamma_l\subset\mathcal D$ with $1\le l\le N-3$ and $q={2(N-l)\over N-l-2} .$ We prove that, under some symmetry assumptions, the number of sign changing solutions to $ (P)_ \epsilon$ increases as $\epsilon$ goes to zero.

2014, 34(11): 4689-4717
doi: 10.3934/dcds.2014.34.4689

*+*[Abstract](387)*+*[PDF](483.2KB)**Abstract:**

In this paper we study extreme events for random walks on homogeneous spaces. We consider the following three cases. On the torus we study closest returns of a random walk to a fixed point in the space. For a random walk on the space of unimodular lattices we study extreme values for lengths of the shortest vector in a lattice. For a random walk on a homogeneous space we study the maximal distance a random walk gets away from an arbitrary fixed point in the space. We prove an exact limiting distribution on the torus and upper and lower bounds for sparse subsequences of random walks in the two other cases. In all three settings we obtain a logarithm law.

2014, 34(11): 4719-4733
doi: 10.3934/dcds.2014.34.4719

*+*[Abstract](348)*+*[PDF](472.8KB)**Abstract:**

We prove several Liouville type results for stationary solutions of the $d$-dimensional compressible Navier-Stokes equations. In particular, we show that when the dimension $d ≥ 4$, the natural requirements $\rho \in L^{\infty} ( \mathbb{R}^d )$, $v \in \dot{H}^1 (\mathbb{R}^d)$ suffice to guarantee that the solution is trivial. For dimensions $d=2,3$, we assume the extra condition $v \in L^{\frac{3d}{d-1}}(\mathbb R^d)$. This improves a recent result of Chae [1].

2014, 34(11): 4735-4749
doi: 10.3934/dcds.2014.34.4735

*+*[Abstract](376)*+*[PDF](429.1KB)**Abstract:**

In this paper we consider the existence and stability of traveling wave solutions to Cauchy problem of diagonalizable quasilinear hyperbolic systems. Under the appropriate small oscillation assumptions on the initial traveling waves, we derive the stability result of the traveling wave solutions, especially for intermediate traveling waves. As the important examples, we will apply the results to some systems arising in fluid dynamics and elementary particle physics.

2014, 34(11): 4751-4764
doi: 10.3934/dcds.2014.34.4751

*+*[Abstract](351)*+*[PDF](387.2KB)**Abstract:**

In the present paper we want to focus on the dichotomy of the non-normal numbers -- on the one hand they are a set of measure zero and on the other hand they are residual -- for dynamical system fulfilling the specification property. These dynamical systems are motivated by $\beta$-expansions. We consider the limiting frequencies of digits in the words of the languagse arising from these dynamical systems, and show that not only a typical $x$ in the sense of Baire is non-normal, but also its Cesàro variants diverge.

2014, 34(11): 4765-4780
doi: 10.3934/dcds.2014.34.4765

*+*[Abstract](420)*+*[PDF](502.0KB)**Abstract:**

Central to the study of $\mathbb{Z}$ actions on compact metric spaces is the $\omega$-limit set, the set of all limit points of a forward orbit. A closed set $K$ is

*internally chain transitive*provided for every $x,y\in K$ there is an $\epsilon$-pseudo-orbit of points from $K$ that starts with $x$ and ends with $y$. It is known in several settings that the property of internal chain transitivity characterizes $\omega$-limit sets. In this paper, we consider actions of $\mathbb{Z}^d$ on compact metric spaces. We give a general definition for shadowing and limit sets in this setting. We characterize limit sets in terms of a more general internal property which we call

*internal mesh transitivity*.

2014, 34(11): 4781-4806
doi: 10.3934/dcds.2014.34.4781

*+*[Abstract](369)*+*[PDF](495.2KB)**Abstract:**

We are interested in the asymptotic behaviour of the number of self-intersections of a trajectory of a Lorentz process in a $\mathbb Z^2$-periodic planar domain with strictly convex obstacles and with finite horizon. We give precise estimates for its expectation and its variance. As a consequence, we establish the almost sure convergence of the self-intersections with a suitable normalization.

2014, 34(11): 4807-4826
doi: 10.3934/dcds.2014.34.4807

*+*[Abstract](402)*+*[PDF](443.1KB)**Abstract:**

This paper is concerned with the bifurcation of limit cycles from a quadratic reversible Lotka-Volterra system with two centers of genus one under small quadratic perturbations. It shows that the cyclicities of each period annulus and two period annuli of the considered system under small quadratic perturbations are two, respectively. This not only gives at least partially a positive answer to an open conjecture, but also improves the corresponding results in the literature. In addition, we present the configurations of limit cycles of the perturbed system as (2, 0), (1, 1), (1, 0), (0, 2), (0, 1) and (0, 0), where $(i,\, j)$ indicates that the perturbed system has $i$ limit cycles surrounding the positive singularity while it has $j$ limit cycles surrounding the negative one.

2014, 34(11): 4827-4854
doi: 10.3934/dcds.2014.34.4827

*+*[Abstract](357)*+*[PDF](734.2KB)**Abstract:**

This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations $\mathrm{LVE}_{\psi}^k$ of a generic autonomous system along a particular solution $\psi$. The main result of this paper is a compact yet explicit and computationally amenable form for said variational systems and their monodromy matrices. Alternatively, the same methods are useful to retrieve, and sometimes simplify, systems satisfied by the coefficients of the Taylor expansion of a formal first integral for a given dynamical system. This is done in preparation for further results within Ziglin-Morales-Ramis theory, specifically those of a constructive nature.

2014, 34(11): 4855-4874
doi: 10.3934/dcds.2014.34.4855

*+*[Abstract](379)*+*[PDF](478.1KB)**Abstract:**

We investigate substitution subshifts and tiling dynamical systems arising from the substitutions (1) $\theta: 0 \rightarrow 001, 1 \rightarrow 11001$ and (2) $\eta: 0 \rightarrow 001, 1 \rightarrow 11100$. We show that the substitution subshifts arising from $\theta$ and $\eta$ have minimal self-joinings and are mildly mixing. We also give a criterion for 1-dimensional tiling systems arising from $\theta$ or $\eta$ to have minimal self-joinings. We apply this to obtain examples of mildly mixing 1-dimensional tiling systems.

2014, 34(11): 4875-4895
doi: 10.3934/dcds.2014.34.4875

*+*[Abstract](299)*+*[PDF](2930.5KB)**Abstract:**

Given a Riemannian manifold $(M,g)$ and a geodesic $\gamma$, the perpendicular part of the derivative of the geodesic flow $\phi_g^t: SM \rightarrow SM$ along $\gamma$ is a linear symplectic map. The present paper gives a new proof of the following Franks' lemma, originally found in [7] and [6]: this map can be perturbed freely within a neighborhood in $Sp(n)$ by a $C^2$-small perturbation of the metric $g$ that keeps $\gamma$ a geodesic for the new metric. Moreover, the size of these perturbations is uniform over fixed length geodesics on the manifold. When $\dim M \geq 3$, the original metric must belong to a $C^2$--open and dense subset of metrics.

2014, 34(11): 4897-4910
doi: 10.3934/dcds.2014.34.4897

*+*[Abstract](371)*+*[PDF](391.5KB)**Abstract:**

We consider the Zakharov-Kuznestov (ZK) equation posed in a limited domain $\mathcal{M}=(0,1)_{x}\times(-\pi /2, \pi /2)^d,$ $ d=1,2$ supplemented with suitable boundary conditions. We prove that there exists a solution $u \in \mathcal C ([0, T]; H^1(\mathcal{M})) $ to the initial and boundary value problem for the ZK equation in both dimensions $2$ and $3$ for every $T>0$. To the best of our knowledge, this is the first result of the global existence of strong solutions for the ZK equation in $3D$.

More importantly, the idea behind the application of anisotropic estimation to cancel the nonlinear term, we believe, is not only suited for this model but can also be applied to other nonlinear equations with similar structures.

At the same time, the uniqueness of solutions is still open in $2D$ and $3D$ due to the partially hyperbolic feature of the model.

2014, 34(11): 4911-4946
doi: 10.3934/dcds.2014.34.4911

*+*[Abstract](576)*+*[PDF](676.9KB)**Abstract:**

In this paper, we continue to study a general nonlocal gradient Patlak-Keller-Segel chemotaxis model in a one dimensional spatial domain. By utilizing the properties of the nonlocal gradient, we first apply the well-known Moser-Alikakos iteration technique plus the heat semigroup theory to obtain the boundedness and hence the global existence of its solution. Then we study the asymptotic behavior of the time-dependent solution, and obtain the limiting equations when the sampling radius $\rho\rightarrow 0$ as well as convergence results when time $t\rightarrow \infty$. Along this way, a ``global" stability issue of the spiky stationary solution for the minimal model is formulated. Finally and importantly, we study the stability of the nonconstant bifurcating solutions. Interestingly, the small size of the cells enhances the occurrence of pattern formation, the stability results are independent of the net creation rate of the chemical, and the stability is closely related to the cell radius $\rho$. Typically, when the cell (net) degradation rate lies below a threshold (stabilizing) value, the cell is stable. Surprisingly, this threshold value is an increasing function of the cell radius. The large cells can compensate their degradation of the chemical signal, and become stable; however, for small cells to be stable, their degradation rate must be less than a threshold value.

2014, 34(11): 4947-4966
doi: 10.3934/dcds.2014.34.4947

*+*[Abstract](441)*+*[PDF](480.3KB)**Abstract:**

In this paper, we study the nonexistence of positive solutions for the following elliptic equation $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u=f(u) & in \quad \mathbb{R}_+^N, \displaystyle \\ \frac{\partial u}{\partial \nu}=g(u) & on \quad \partial \mathbb{R}_+^N \end{array} \right. $$ and elliptic system $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u_1=f_1(u_1,u_2) &in \quad \mathbb{R}_+^N, \\ \\-\Delta u_2=f_2(u_1,u_2) & in\quad \mathbb{R}_+^N, \\ \displaystyle \\ \frac{\partial u_1}{\partial \nu}=g_1(u_1,u_2),\quad \frac{\partial u_2}{\partial \nu}=g_2(u_1,u_2) & on \quad \partial \mathbb{R}_+^N. \end{array} \right. $$ We will prove that these problems possess no positive solutions under some assumptions on nonlinear terms. The main technique we use is the moving plane method in an integral form.

2014, 34(11): 4967-4986
doi: 10.3934/dcds.2014.34.4967

*+*[Abstract](441)*+*[PDF](533.3KB)**Abstract:**

This paper deals with the Cauchy problem for a generalized $b$-equation with higher-order nonlinearities $y_{t}+u^{m+1}y_{x}+bu^{m}u_{x}y=0$, where $b$ is a constant and $m\in\mathbb{N}$, the notation $y:= (1-\partial_x^2) u$, which includes the famous $b$-equation and Novikov equation as special cases. The local well-posedness in critical Besov space $B^{3/2}_{2,1}$ is established. Moreover, a lower bound for the maximal existence time is derived. Finally, the persistence properties in weighted $L^p$ spaces for the solution of this equation are considered, which extend the work of Brandolese [L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Not. 22 (2012), 5161-5181] on persistence properties to more general equation with higher-order nonlinearities.

2014, 34(11): 4987-4987
doi: 10.3934/dcds.2014.34.4987

*+*[Abstract](348)*+*[PDF](189.2KB)**Abstract:**

N/A

2014, 34(11): 4989-4995
doi: 10.3934/dcds.2014.34.4989

*+*[Abstract](582)*+*[PDF](377.8KB)**Abstract:**

We provide a corrected proof of [4, Theorem 2.2], which preserves the validity of the theorem exactly under those assumptions as stated in the original paper.

2017 Impact Factor: 1.179

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