# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

March 2015 , Volume 35 , Issue 3

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2015, 35(3): 793-806 doi: 10.3934/dcds.2015.35.793 +[Abstract](2355) +[PDF](443.2KB)
Abstract:
We give an example of a sequential dynamical system consisting of intermittent-type maps which exhibits loss of memory with a polynomial rate of decay. A uniform bound holds for the upper rate of memory loss. The maps may be chosen in any sequence, and the bound holds for all compositions.
2015, 35(3): 807-827 doi: 10.3934/dcds.2015.35.807 +[Abstract](2023) +[PDF](2070.7KB)
Abstract:
Fundamental global similarity solutions of the standard form $$u_\gamma(x,t) = t^{-\alpha_\gamma} f_\gamma(y),\,\,\mbox{with the rescaled variable}\,\,\, y= x/{t^{\beta_\gamma}}, \,\, \beta_\gamma= \frac {1-n \alpha_\gamma}{10},$$ where $\alpha_\gamma>0$ are real nonlinear eigenvalues ($\gamma$ is a multiindex in $\mathbb{R}^N$) of the tenth-order thin film equation (TFE-10) \begin{eqnarray*} \label{i1a} u_{t} = \nabla \cdot (|u|^{n} \nabla \Delta^4 u) \quad in \quad \mathbb{R}^N \times \mathbb{R}_+ \,, \quad n>0,                   (0.1) \end{eqnarray*} are studied. The present paper continues the study began in [1]. Thus, the following questions are also under scrutiny:
(I) Further study of the limit $n \to 0$, where the behaviour of finite interfaces and solutions as $y \to \infty$ are described. In particular, for $N=1$, the interfaces are shown to diverge as follows: $$|x_0(t)| \sim 10 ( \frac{1}{n}\sec ( \frac{4\pi}{9} ) )^{\frac 9{10}} t^{\frac 1{10}} \to \infty as n \to 0^+.$$
(II) For a fixed $n \in (0, \frac 98)$, oscillatory structures of solutions near interfaces.
(III) Again, for a fixed $n \in (0, \frac 98)$, global structures of some nonlinear eigenfunctions $\{f_\gamma\}_{|\gamma| \ge 0}$ by a combination of numerical and analytical methods.
2015, 35(3): 829-855 doi: 10.3934/dcds.2015.35.829 +[Abstract](1590) +[PDF](635.4KB)
Abstract:
In this paper we give a criterion for a special flow to be not isomorphic to its inverse which is a refine of a result in [6]. We apply this criterion to special flows $T^f$ built over ergodic interval exchange transformations $T:[0,1)\to[0,1)$ (IETs) and under piecewise absolutely continuous roof functions $f:[0,1)\to\mathbb{R}_+$. We show that for almost every IET $T$ if $f$ is absolutely continuous over exchanged intervals and has non-zero sum of jumps then the special flow $T^f$ is not isomorphic to its inverse. The same conclusion is valid for a typical piecewise constant roof function.
2015, 35(3): 857-870 doi: 10.3934/dcds.2015.35.857 +[Abstract](1714) +[PDF](432.8KB)
Abstract:
Nash and Sobolev inequalities are known to be equivalent to ultracontractive properties of heat-like Markov semigroups, hence to uniform on-diagonal bounds on their kernel densities. In non ultracontractive settings, such bounds can not hold, and (necessarily weaker, non uniform) bounds on the semigroups can be derived by means of weighted Nash (or super-Poincaré) inequalities. The purpose of this note is to show how to check these weighted Nash inequalities in concrete examples of reversible diffusion Markov semigroups in $\mathbb{R}^d$, in a very simple and general manner. We also deduce off-diagonal bounds for the Markov kernels of the semigroups, refining E. B. Davies' original argument.
2015, 35(3): 871-889 doi: 10.3934/dcds.2015.35.871 +[Abstract](2679) +[PDF](457.9KB)
Abstract:
In this paper, we consider the Cauchy problem for a generalized Camassa-Holm equation. We establish the local well-posedness in the Besov space $B_{2,1}^{3/2}$ and also prove that the local well-posedness fails in the Besov space $B_{2,\infty}^{3/2}$.
2015, 35(3): 891-907 doi: 10.3934/dcds.2015.35.891 +[Abstract](1781) +[PDF](467.7KB)
Abstract:
We consider the semilinear heat equation $$\label{problemAbstract}\left\{\begin{array}{ll}v_t-\Delta v= |v|^{p-1}v & \mbox{ in }\Omega\times (0,T)\\ v=0 & \mbox{ on }\partial \Omega\times (0,T)\\ v(0)=v_0 & \mbox{ in }\Omega \end{array}\right.\tag{\mathcal P_p}$$ where $p>1$, $\Omega$ is a smooth bounded domain of $\mathbb{R}^2$, $T\in (0,+\infty]$ and $v_0$ belongs to a suitable space. We give general conditions for a family $u_p$ of sign-changing stationary solutions of ($\mathcal P_p$), under which the solution of ($\mathcal P_p$) with initial value $v_0=\lambda u_p$ blows up in finite time if $|\lambda-1|>0$ is sufficiently small and $p$ is sufficiently large. Since for $\lambda=1$ the solution is global, this shows that, in general, the set of the initial conditions for which the solution is global is not star-shaped with respect to the origin. In [5] this phenomenon has been previously observed in the case when the domain is a ball and the sign changing stationary solution is radially symmetric. Our conditions are more general and we provide examples of stationary solutions $u_p$ which are not radial and exhibit the same behavior.
2015, 35(3): 909-916 doi: 10.3934/dcds.2015.35.909 +[Abstract](2328) +[PDF](343.3KB)
Abstract:
This paper addresses the hydrodynamical stability of nonlinear geophysical equatorial waves in the $f-$plane approximation. By implementing the short-wavelength perturbation approach, we show that certain westward-propagating equatorial waves are linearly unstable when the wave steepness exceeds a specific threshold.
2015, 35(3): 917-934 doi: 10.3934/dcds.2015.35.917 +[Abstract](2654) +[PDF](430.0KB)
Abstract:
The initial-boundary value problem for the equations of compressible viscoelastic flows is considered in a bounded domain of three-dimensional spatial dimensions. The global existence of strong solution near equilibrium is established. Uniform estimates in $W^{1,q}$ with $q>3$ on the density and deformation gradient are also obtained.
2015, 35(3): 935-942 doi: 10.3934/dcds.2015.35.935 +[Abstract](2555) +[PDF](359.9KB)
Abstract:
In this paper, we study the best constant of the following discrete Hardy-Littlewood-Sobolev inequality, $$\sum_{i,j,i\neq j}\frac{f_{i}g_{j}}{\mid i-j\mid^{n-\alpha}}\leq C_{r,s,\alpha} |f|_{l^r} |g|_{l^s},$$where $i,j\in \mathbb Z^n$, $r,s>1$, $0 < \alpha < n$, and $\frac {1} {r} + \frac {1} {s} + \frac {n-\alpha}{n} \geq 2$. Indeed, we prove that the best constant is attainable in the supercritical case $\frac {1}{r} + \frac {1} {s} + \frac {n-\alpha}{n} > 2$.
2015, 35(3): 943-966 doi: 10.3934/dcds.2015.35.943 +[Abstract](2377) +[PDF](492.6KB)
Abstract:
In this paper, we study the existence of ground state solutions to the nonlinear Kirchhoff type equations $- m \left( \| \nabla u \|_{L^2(\mathbf{R}^N)}^{2} \right) \Delta u + V(x) u = |u|^{p-1} u \quad {\rm in}\ \mathbf{R}^N, \ u \in H^1(\mathbf{R}^N), \ N \geq 1$ where $1 < p < \infty$ when $N=1,2$, $1 < p < (N+2)/(N-2)$ when $N \geq 3$, $m: [0,\infty) \to (0,\infty)$ is a continuous function and $V:\mathbf{R}^N \to \mathbf{R}$ a smooth function. Under suitable conditions on $m(s)$ and $V$, it is shown that a ground state solution to the above equation exists.
2015, 35(3): 967-983 doi: 10.3934/dcds.2015.35.967 +[Abstract](1900) +[PDF](394.2KB)
Abstract:
Under the condition that unstable manifolds are one dimensional, the derivative formula of the potential function of the generalized SRB measure with respect to the underlying dynamical system is extended from the hyperbolic attractor case to the general case when the hyperbolic set intersecting with unstable manifolds is a Cantor set. It leads to derivative formulas of objects and quantities that characterize a uniformly hyperbolic system, including the generalized SBR measure and its entropy, the root of the Bowen's equation, and the Hausdorff dimension of the hyperbolic set on a dimension two Riemannian manifold.
2015, 35(3): 985-1008 doi: 10.3934/dcds.2015.35.985 +[Abstract](2756) +[PDF](599.2KB)
Abstract:
This paper is concerned with well-posedness and asymptotic behavior to a class of extensible beams with nonlinear fractional damping and source terms $$u_{tt} + \Delta^2 u - M\left(\int_{\Omega}|\nabla u|^2 \, dx \right)\Delta u + N\left(\int_{\Omega}|\nabla u|^2 \, dx \right) \left(-\Delta\right)^{\theta} u_t + f(u) = h$$ in $\Omega\times(0,\infty)$, where $\Omega\subset \mathbb{R}^q$ is a bounded domain with smooth boundary $\partial\Omega$, $0\leq \theta \le 1$, $M$ and $N$ are scalar functions, $f(u)$ is a source term and $h$ is a forcing term. When $\theta=1$ we have the classical strong damping $-\Delta u_t$, and if $\theta=0$ then weak damping $u_t$ is considered. Under the mathematical point of view the abstract setting of $\left(-\Delta\right)^{\theta}u_t$ with $0< \theta < 1$ constitutes a way to change the characteristic of the problem in wave and plate vibrations, and maybe the decay rate of solutions mainly when we consider nonconstant coefficient damping $N$. The main purpose in the present paper is to show that the transition from the case $0< \theta \leq 1$ to the case $\theta=0$ does not produce any influence on the asymptotic behavior of solutions, that is, all results are obtained by moving $\theta\in[0,1]$ uniformly. Our main result establishes the existence of finite-dimensional compact global and exponential attractors by using an approach on quasi-stable dynamical systems given by Chueshov and Lasiecka (2010).
2015, 35(3): 1009-1037 doi: 10.3934/dcds.2015.35.1009 +[Abstract](2230) +[PDF](719.6KB)
Abstract:
In this work we study the semilinear wave equation of the form $u_{tt}=u_{xx} + {\lambda}/{ (1-u)^2},$ with homogeneous Dirichlet boundary conditions and suitable initial conditions, which, under appropriate circumstances, serves as a model of an idealized electrostatically actuated MEMS device. First we establish local existence of the solutions of the problem for any $\lambda>0.$ Then we focus on the singular behaviour of the solution, which occurs through finite-time quenching, i.e. when $||u(\cdot,t)||_{\infty}\to 1$ as $t\to t^*- < \infty$, investigating both conditions for quenching and the quenching profile of $u.$ To this end, the non-existence of a regular similarity solution near a quenching point is first shown and then a formal asymptotic expansion is used to determine the local form of the solution. Finally, using a finite difference scheme, we solve the problem numerically, illustrating the preceding results.
2015, 35(3): 1039-1057 doi: 10.3934/dcds.2015.35.1039 +[Abstract](2398) +[PDF](440.8KB)
Abstract:
This paper is concerned with the integral system $$\left \{ \begin{array}{ll} &u(x)=\int_{R^n}\frac{|x-y|^\lambda dy}{v^q(y)},\quad u>0~in~R^n,\\ &v(x)=\int_{R^n}\frac{|x-y|^\lambda dy}{u^p(y)},\quad v>0~in~R^n, \end{array} \right.$$ where $n \geq 1$, $p,q,\lambda \neq 0$. Such an integral system appears in the study of the conformal geometry. We obtain several necessary conditions for the existence of the $C^1$ positive entire solutions, particularly including the critical condition $$\frac{1}{p-1}+\frac{1}{q-1}=\frac{\lambda}{n},$$ which is the necessary and sufficient condition for the invariant of the system and some energy functionals under the scaling transformation. The necessary condition $\frac{1}{p-1}+\frac{1}{q-1}=\frac{\lambda}{n}$ can be relaxed to another weaker one $\min\{p,q\}>\frac{n+\lambda}{\lambda}$ for the system with double bounded coefficients. In addition, we classify the radial solutions in the case of $p=q$ as the form $$u(x)=v(x)=a(b^2+|x-x_0|^2)^{\frac{\lambda}{2}}$$ with $a,b>0$ and $x_0 \in R^n$. Finally, we also deduce some analogous necessary conditions of existence for the weighted system.
2015, 35(3): 1059-1073 doi: 10.3934/dcds.2015.35.1059 +[Abstract](2117) +[PDF](446.7KB)
Abstract:
A topological dynamical system induces two natural systems, one is on the hyperspace and the other one is on the probability measures space. The connection among some dynamical properties on the original space and on the induced spaces are investigated. Particularly, a minimal weakly mixing system which induces a $P$-system on the probability measures space is constructed and some disjointness result is obtained.
2015, 35(3): 1075-1090 doi: 10.3934/dcds.2015.35.1075 +[Abstract](2251) +[PDF](403.4KB)
Abstract:
In general the center--focus problem cannot be solved, but in the case that the singularity has purely imaginary eigenvalues there are algorithms to solving it. The present paper implements one of these algorithms for the polynomial differential systems of the form $\dot x= -y + x f(x) g(y),\quad \dot y= x+y f(x) g(y),$ where $f(x)$ and $g(y)$ are arbitrary polynomials. These differential systems have constant angular speed and are also called rigid systems. More precisely, in this paper we give the center conditions for these systems, i.e. the necessary and sufficient conditions in order that they have an uniform isochronous center. In particular, the existence of a focus with the highest order is also studied.
2015, 35(3): 1091-1102 doi: 10.3934/dcds.2015.35.1091 +[Abstract](2011) +[PDF](371.7KB)
Abstract:
It is well known that invariant algebraic curves of polynomial differential systems play an important role in questions regarding integrability of these systems. But do they also have a role in relation to limit cycles? In this article we show that not only they do have a role in the production of limit cycles in polynomial perturbations of such systems but that algebraic invariant curves can even generate algebraic limit cycles in such perturbations. We prove that when we perturb any quadratic system with an invariant ellipse surrounding a center (quadratic systems with center always have invariant algebraic curves and some of them have invariant ellipses) within the class of quadratic differential systems, there is at least one 1-parameter family of such systems having a limit cycle bifurcating from the ellipse. Therefore the cyclicity of the period annulus of such systems is at least one.
2015, 35(3): 1103-1138 doi: 10.3934/dcds.2015.35.1103 +[Abstract](1752) +[PDF](534.3KB)
Abstract:
The main objective of this article is to postulate a principle of interaction dynamics (PID) and to derive field equations coupling the four fundamental interactions based on first principles. PID is a least action principle subject to div$_A$-free constraints for the variational element with $A$ being gauge potentials. The Lagrangian action is uniquely determined by 1) the principle of general relativity, 2) the $U(1)$, $SU(2)$ and $SU(3)$ gauge invariances, 3) the Lorentz invariance, and 4) the principle of representation invariance (PRI), introduced in [11]. The unified field equations are then derived using PID. The field model spontaneously breaks the gauge symmetries, and gives rise to a new mass generation mechanism. The unified field model introduces a natural duality between the mediators and their dual mediators, and can be easily decoupled to study each individual interaction when other interactions are negligible. The unified field model, together with PRI and PID applied to individual interactions, provides clear explanations and solutions to a number of outstanding challenges in physics and cosmology, including e.g. the dark energy and dark matter phenomena, the quark confinement, asymptotic freedom, short-range nature of both strong and weak interactions, decay mechanism of sub-atomic particles, baryon asymmetry, and the solar neutrino problem.
2015, 35(3): 1139-1162 doi: 10.3934/dcds.2015.35.1139 +[Abstract](1969) +[PDF](495.8KB)
Abstract:
We apply KAM theory to the equation of the forced relativistic pendulum to prove that all the solutions have bounded momentum. Subsequently, we detect the existence of quasiperiodic solutions in a generalized sense. This is achieved using a modified version of the Aubry-Mather theory for compositions of twist maps.
2015, 35(3): 1163-1177 doi: 10.3934/dcds.2015.35.1163 +[Abstract](1860) +[PDF](393.6KB)
Abstract:
In the set of continuously differentiable maps of the real line with a discontinuity, equipped with the uniform topology, it will be shown that the subset of transitive ones has nonempty interior.
2015, 35(3): 1179-1192 doi: 10.3934/dcds.2015.35.1179 +[Abstract](2278) +[PDF](326.4KB)
Abstract:
In this paper we consider a wave equation with a viscoelastic boundary damping localized on a part of the boundary. We establish an explicit and general decay rate result that allows a larger class of relaxation functions and generalizes previous results existing in the literature.
2015, 35(3): 1193-1230 doi: 10.3934/dcds.2015.35.1193 +[Abstract](1786) +[PDF](695.7KB)
Abstract:
We construct and investigate local invariant manifolds for a large class of quasilinear parabolic problems with fully nonlinear dynamical boundary conditions and study their attractivity properties. In a companion paper we have developed the corresponding solution theory. Examples for the class of systems considered are reaction--diffusion systems or phase field models with dynamical boundary conditions and to the two--phase Stefan problem with surface tension.
2015, 35(3): 1231-1238 doi: 10.3934/dcds.2015.35.1231 +[Abstract](2290) +[PDF](346.5KB)
Abstract:
In this paper, we first discuss minimal sets in the compact dynamical system with weak specification property. On the basis of this discussion, we show that an expansive system with weak specification property displays a stronger form of $\omega$-chaos.
2015, 35(3): 1239-1284 doi: 10.3934/dcds.2015.35.1239 +[Abstract](3099) +[PDF](966.1KB)
Abstract:
We study a diffusive Lotka-Volterra competition system with advection under Neumann boundary conditions. Our system models a competition relationship that one species escape from the region of high population density of their competitors in order to avoid competition. We establish the global existence of bounded classical solutions to the system over one-dimensional finite domains. For multi-dimensional domains, globally bounded classical solutions are obtained for a parabolic-elliptic system under proper assumptions on the system parameters. These global existence results make it possible to study bounded steady states in order to model species segregation phenomenon. We then investigate the one-dimensional stationary problem. Through bifurcation theory, we obtain the existence of nonconstant positive steady states, which are small perturbations from the positive equilibrium; we also rigourously study the stability of these bifurcating solutions when diffusion coefficients of the escaper and its competitor are large and small respectively. In the limit of large advection rate, we show that the reaction-advection-diffusion system converges to a shadow system involving the competitor population density and an unknown positive constant. Existence and stability of positive nonconstant solutions to the shadow system have also been obtained through bifurcation theories. Finally, we construct infinitely many single interior transition layers to the shadow system when crowding rate of the escapers and diffusion rate of their interspecific competitors are sufficiently small. The transition-layer solutions can be used to model the interspecific segregation phenomenon.
2015, 35(3): 1285-1326 doi: 10.3934/dcds.2015.35.1285 +[Abstract](2358) +[PDF](571.3KB)
Abstract:
Random dynamical systems with countably many maps which admit countable Markov partitions on complete metric spaces such that the resulting Markov systems are uniformly continuous and contractive are considered. A non-degeneracy and a consistency conditions for such systems, which admit some proper Markov partitions of connected spaces, are introduced, and further sufficient conditions for them are provided. It is shown that every uniformly continuous Markov system associated with a continuous random dynamical system is consistent if it has a dominating Markov chain. A necessary and sufficient condition for the existence of an invariant Borel probability measure for such a non-degenerate system with a dominating Markov chain and a finite (16) is given. The condition is also sufficient if the non-degeneracy is weakened with the consistency condition. A further sufficient condition for the existence of an invariant measure for such a consistent system which involves only the properties of the dominating Markov chain is provided. In particular, it implies that every such a consistent system with a finite Markov partition and a finite (16) has an invariant Borel probability measure. A bijective map between these measures and equilibrium states associated with such a system is established in the non-degenerate case. Some properties of the map and the measures are given.
2015, 35(3): 1327-1358 doi: 10.3934/dcds.2015.35.1327 +[Abstract](2114) +[PDF](532.6KB)
Abstract:
This paper is concerned with the the initial value problem for higher dimensional Camassa-Holm equations. Firstly, the local well-posedness for this equations in both supercritical and critical Besov spaces are established. Then two blow-up criterions of strong solutions to the equations are derived. Finally, the analyticity of its solutions is proved in both variables, globally in space and locally in time.
2015, 35(3): 1359-1385 doi: 10.3934/dcds.2015.35.1359 +[Abstract](2034) +[PDF](576.9KB)
Abstract:
In this paper we consider the equations of the unsteady viscous, incompressible, and heat conducting magnetohydrodynamic flows in a bounded three-dimensional domain with Lipschitz boundary. By an approximation scheme and a weak convergence method, the existence of a weak solution to the three-dimensional density dependent generalized incompressible magnetohydrodynamic equations with large data is obtained.
2015, 35(3): 1387-1390 doi: 10.3934/dcds.2015.35.1387 +[Abstract](1881) +[PDF](284.5KB)
Abstract:
We amend some notations and assumptions in [1] and correct a flaw in the proof of [[1], Lemma 2.4].

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