# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

April 2016 , Volume 36 , Issue 4

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2016, 36(4): 1759-1788 doi: 10.3934/dcds.2016.36.1759 +[Abstract](921) +[PDF](557.2KB)
Abstract:
In this paper, we obtain sharp estimates of fully bubbling solutions of the $B_2$ Toda system in a compact Riemann surface. Our main goal in this paper are (i) to obtain sharp convergence rate, (ii) to completely determine the location of bubbles, (iii) to derive the $\partial_z^2$ condition.
2016, 36(4): 1789-1811 doi: 10.3934/dcds.2016.36.1789 +[Abstract](867) +[PDF](496.4KB)
Abstract:
In this paper we establish existence and stability results concerning fully nontrivial solitary-wave solutions to 3-coupled nonlinear Schrödinger system \begin{equation*} i\partial_t u_{j}+ \partial_{xx}u_{j}+ \left(\sum_{k=1}^{3} a_{kj} |u_k|^{p}\right)|u_j|^{p-2}u_j = 0, \ j=1,2,3, \end{equation*} where $u_j$ are complex-valued functions of $(x,t)\in \mathbb{R}^{2}$ and $a_{kj}$ are positive constants satisfying $a_{kj}=a_{jk}$ (symmetric attractive case). Our approach improves many of the previously known results. In all variational methods used previously to study the stability of solitary waves, which we are aware of, the constraint functionals were not independently chosen. Here we study a problem of minimizing the energy functional subject to three independent $L^2$ mass constraints and establish existence and stability results for a true three-parameter family of solitary waves.
2016, 36(4): 1813-1845 doi: 10.3934/dcds.2016.36.1813 +[Abstract](1055) +[PDF](618.9KB)
Abstract:
By virtue of $\Gamma-$convergence arguments, we investigate the stability of variational eigenvalues associated with a given topological index for the fractional $p-$Laplacian operator, in the singular limit as the nonlocal operator converges to the $p-$Laplacian. We also obtain the convergence of the corresponding normalized eigenfunctions in a suitable fractional norm.
2016, 36(4): 1847-1868 doi: 10.3934/dcds.2016.36.1847 +[Abstract](916) +[PDF](432.1KB)
Abstract:
In this paper, the compressible magnetohydrodynamic system with some smallness and symmetry assumptions on the time periodic external force is considered in $\mathbb{R}^3$. Based on the uniform estimates and the topological degree theory, we prove the existence of a time periodic solution in a bounded domain. Then by a limiting process, the result in the whole space $\mathbb{R}^3$ is obtained.
2016, 36(4): 1869-1880 doi: 10.3934/dcds.2016.36.1869 +[Abstract](740) +[PDF](415.4KB)
Abstract:
We consider semilinear equations of the form $p(D)u=F(u)$, with a locally bounded nonlinearity $F(u)$, and a linear part $p(D)$ given by a Fourier multiplier. The multiplier $p(\xi)$ is the sum of positively homogeneous terms, with at least one of them non smooth. This general class of equations includes most physical models for traveling waves in hydrodynamics, the Benjamin-Ono equation being a basic example.
We prove sharp pointwise decay estimates for the solutions to such equations, depending on the degree of the non smooth terms in $p(\xi)$. When the nonlinearity is smooth we prove similar estimates for the derivatives of the solution, as well as holomorphic extension to a strip, for analytic nonlinearity.
2016, 36(4): 1881-1903 doi: 10.3934/dcds.2016.36.1881 +[Abstract](778) +[PDF](499.6KB)
Abstract:
Let $\alpha\in(0,1)$, $\Omega$ be a bounded open domain in $\mathbb{R}^N$ ($N\ge 2$) with $C^2$ boundary $\partial\Omega$ and $\omega$ be the Hausdorff measure on $\partial\Omega$. We denote by $\frac{\partial^\alpha \omega}{\partial \vec{n}^\alpha}$ a measure $$\langle\frac{\partial^\alpha \omega}{\partial \vec{n}^\alpha},f\rangle=\int_{\partial\Omega}\frac{\partial^\alpha f(x)}{\partial \vec{n}_x^\alpha} d\omega(x),\quad f\in C^1(\bar\Omega),$$ where $\vec{n}_x$ is the unit outward normal vector at point $x\in\partial\Omega$. In this paper, we prove that problem $$\label{0.1} \begin{array}{lll} (-\Delta)^\alpha u+g(u)=k\frac{\partial^\alpha \omega}{\partial \vec{n}^\alpha}\quad & {\rm in}\quad \bar\Omega, (1)\\ \phantom{ (-\Delta)^\alpha +g(u)} u=0\quad & {\rm in}\quad \bar\Omega^c \end{array}$$ admits a unique weak solution $u_k$ under the hypotheses that $k>0$, $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$ and $g$ is a nondecreasing function satisfying extra conditions. We prove that the weak solution of (1) is a classical solution of $$\begin{array}{lll} \ \ \ (-\Delta)^\alpha u+g(u)=0\quad & {\rm in}\quad \Omega,\\ \phantom{------\ } \ \ \ u=0\quad & {\rm in}\quad \mathbb{R}^N\setminus\bar\Omega,\\ \phantom{} \lim_{x\in\Omega,x\to\partial\Omega}u(x)=+\infty. \end{array}$$
2016, 36(4): 1905-1926 doi: 10.3934/dcds.2016.36.1905 +[Abstract](916) +[PDF](579.0KB)
Abstract:
In this paper we obtain some new inhomogeneous Strichartz estimates for the fractional Schrödinger equation in the radial case. Then we apply them to the well-posedness theory for the equation $i\partial_{t}u+|\nabla|^{\alpha}u=V(x,t)u$, $1<\alpha<2$, with radial $\dot{H}^\gamma$ initial data below $L^2$ and radial potentials $V\in L_t^rL_x^w$ under the scaling-critical range $\alpha/r+n/w=\alpha$.
2016, 36(4): 1927-1955 doi: 10.3934/dcds.2016.36.1927 +[Abstract](766) +[PDF](558.9KB)
Abstract:
In this paper we investigate a basic one-dimensional viscous gas-liquid model based on the two-fluid model formulation. The gas is modeled as a polytropic gas whereas liquid is assumed to be incompressible. A main challenge with this model is the appearance of a non-conservative pressure term which possibly also blows up at transition to single-phase liquid flow (due to incompressible liquid). We investigate the model both in a finite domain (initial-boundary value problem) and in the whole space (Cauchy problem). We demonstrate that under appropriate smallness conditions on initial data we can obtain time-independent estimates which allow us to show existence and uniqueness of regular solutions as well as to gain insight into the long-time behavior of the model. These results rely strongly on the fact that we can derive appropriate upper and lower uniform bounds on the gas and liquid mass. In particular, the estimates guarantee that gas does not vanish at any point for any time when initial gas phase has a positive lower limit. The discussion of the Cauchy problem is general enough to take into account the possibility that the liquid phase may vanish at some points at initial time.
2016, 36(4): 1957-1982 doi: 10.3934/dcds.2016.36.1957 +[Abstract](853) +[PDF](584.1KB)
Abstract:
We prove that a $C^3$ critical circle map without periodic points has zero Lyapunov exponent with respect to its unique invariant Borel probability measure. Moreover, no critical point of such a map satisfies the Collet-Eckmann condition. This result is proved directly from the well-known real a-priori bounds, without using Pesin's theory. We also show how our methods yield an analogous result for infinitely renormalizable unimodal maps of any combinatorial type. Finally we discuss an application of these facts to the study of neutral measures of certain rational maps of the Riemann sphere.
2016, 36(4): 1983-2025 doi: 10.3934/dcds.2016.36.1983 +[Abstract](777) +[PDF](596.7KB)
Abstract:
Rauzy Classes and Extended Rauzy Classes are equivalence classes of permutations that arise when studying Interval Exchange Transformations. In 2003, Kontsevich and Zorich classified Extended Rauzy Classes by using data from Translation Surfaces, which are associated to IET's thanks to the Zippered Rectangle Construction of Veech from 1982. In 2009, Boissy finalized the classification of Rauzy Classes also using information from Translation Surfaces. We present in this paper specialized moves in (Extended) Rauzy Classes that allow us to prove the sufficiency and necessity in the previous classification theorems. These results provide a complete, and purely combinatorial, proof of these known results. We end with some general statements about our constructed move.
2016, 36(4): 2027-2046 doi: 10.3934/dcds.2016.36.2027 +[Abstract](700) +[PDF](468.2KB)
Abstract:
In this work we extend well-known techniques for solving the Poincaré-Lyapunov nondegenerate analytic center problem in the plane to the 3-dimensional center problem at the zero-Hopf singularity. Thus we characterize the existence of a neighborhood of the singularity completely foliated by periodic orbits (including continua of equilibria) via an analytic Poincaré return map. The vanishing of the first terms in a Taylor expansion of the associated displacement map provides us with the necessary 3-dimensional center conditions in the parameter space of the family whereas the sufficiency is obtained through symmetry-integrability methods. Finally we use the proposed method to classify the 3-dimensional centers of some quadratic polynomial differential families possessing a zero-Hopf singularity.
2016, 36(4): 2047-2067 doi: 10.3934/dcds.2016.36.2047 +[Abstract](715) +[PDF](1085.8KB)
Abstract:
We consider atomic chains with nonlocal particle interactions and prove the existence of near-sonic solitary waves. Both our result and the general proof strategy are reminiscent of the seminal paper by Friesecke and Pego on the KdV limit of chains with nearest neighbor interactions but differ in the following two aspects: First, we allow for a wider class of atomic systems and must hence replace the distance profile by the velocity profile. Second, in the asymptotic analysis we avoid a detailed Fourier pole characterization of the nonlocal integral operators and employ the contraction mapping principle to solve the final fixed point problem.
2016, 36(4): 2069-2084 doi: 10.3934/dcds.2016.36.2069 +[Abstract](745) +[PDF](429.1KB)
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2016, 36(4): 2085-2102 doi: 10.3934/dcds.2016.36.2085 +[Abstract](835) +[PDF](461.5KB)
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We consider finite energy solutions for the damped and driven two-dimensional Navier--Stokes equations in the plane and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension when the forcing term belongs to the whole scale of homogeneous Sobolev spaces from $-1$ to $1$.
2016, 36(4): 2103-2112 doi: 10.3934/dcds.2016.36.2103 +[Abstract](829) +[PDF](507.1KB)
Abstract:
If the Julia set of a quartic polynomial with certain conditions is neither connected nor totally disconnected, there exists a homeomorphism between the set of all components of the filled-in Julia set and some subset of the corresponding symbol space. The question is to determine the quartic polynomials exhibiting such a dynamics and describe the topology of the connected components of their filled-in Julia sets. In this paper, we answer the question, namely we show that for any two quadratic Julia sets, there exists a quartic polynomial whose Julia set includes copies of the two quadratic Julia sets.
2016, 36(4): 2113-2132 doi: 10.3934/dcds.2016.36.2113 +[Abstract](773) +[PDF](457.5KB)
Abstract:
We consider optimal control problems where the running cost of the trajectory is evaluated by a probability measure on $\mathbb{R}_+$. As a particular case, we take the Cesàro average of the running cost over a fixed horizon. The limit of the value with Cesàro average when the horizon tends to infinity is widely studied in the literature. We address the more general question of the existence of a limit for values defined by general evaluations satisfying certain long-term condition.
For this aim, we introduce an asymptotic regularity condition for a sequence of probability measures on $\mathbb{R}_+$. Our main result is that, for any sequence of probability measures on $\mathbb{R}_+$ satisfying this condition, the associated value functions converge uniformly if and only if this family is totally bounded for the uniform norm.
As a byproduct, we obtain the existence of a limit value (for general evaluations) for control systems defined on a compact invariant domain and satisfying suitable nonexpansive property.
2016, 36(4): 2133-2170 doi: 10.3934/dcds.2016.36.2133 +[Abstract](980) +[PDF](5124.7KB)
Abstract:
This article presents stability analytical results of a two component reaction-diffusion system with linear cross-diffusion posed on continuously evolving domains. First the model system is mapped from a continuously evolving domain to a reference stationary frame resulting in a system of partial differential equations with time-dependent coefficients. Second, by employing appropriately asymptotic theory, we derive and prove cross-diffusion-driven instability conditions for the model system for the case of slow, isotropic domain growth. Our analytical results reveal that unlike the restrictive diffusion-driven instability conditions on stationary domains, in the presence of cross-diffusion coupled with domain evolution, it is no longer necessary to enforce cross nor pure kinetic conditions. The restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Reaction-cross-diffusion models with equal diffusion coefficients in the principal components as well as those of the short-range inhibition, long-range activation and activator-activator form can generate patterns only in the presence of cross-diffusion coupled with domain evolution. To confirm our theoretical findings, detailed parameter spaces are exhibited for the special cases of isotropic exponential, linear and logistic growth profiles. In support of our theoretical predictions, we present evolving or moving finite element solutions exhibiting patterns generated by a short-range inhibition, long-range activation reaction-diffusion model with linear cross-diffusion in the presence of domain evolution.
2016, 36(4): 2171-2191 doi: 10.3934/dcds.2016.36.2171 +[Abstract](778) +[PDF](453.5KB)
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In this paper, we are concerned with the Cauchy problem of the new integrable four-component system with cubic nonlinearity. We establish the local well-posedness in a range of the Besov spaces. Then the precise blow-up scenario for strong solutions to the system is derived.
2016, 36(4): 2193-2204 doi: 10.3934/dcds.2016.36.2193 +[Abstract](634) +[PDF](350.1KB)
Abstract:
We consider the Maxwell-Chern-Simons-Higgs system in Lorenz gauge and use a null condition to show local well-psoedness for low regularity data. This improves a recent result of J. Yuan.
2016, 36(4): 2205-2227 doi: 10.3934/dcds.2016.36.2205 +[Abstract](1003) +[PDF](509.5KB)
Abstract:
We consider spike vector solutions for the nonlinear Schrödinger system \begin{equation*} \left\{ \begin{array}{ll} -\varepsilon^{2}\Delta u+P(x)u=\mu u^{3}+\beta v^2u \ \hbox{in}\ \mathbb{R}^3,\\ -\varepsilon^{2}\Delta v+Q(x)v=\nu v^{3} +\beta u^2v \ \ \hbox{in}\ \mathbb{R}^3,\\ u, v >0 \,\ \hbox{in}\ \mathbb{R}^3, \end{array} \right. \end{equation*} where $\varepsilon > 0$ is a small parameter, $P(x)$ and $Q(x)$ are positive potentials, $\mu>0, \nu>0$ are positive constants and $\beta\neq 0$ is a coupling constant. We investigate the effect of potentials and the nonlinear coupling on the solution structure. For any positive integer $k\ge 2$, we construct $k$ interacting spikes concentrating near the local maximum point $x_{0}$ of $P(x)$ and $Q(x)$ when $P(x_{0})=Q(x_{0})$ in the attractive case. In contrast, for any two positive integers $k\ge 2$ and $m\ge 2$, we construct $k$ interacting spikes for $u$ near the local maximum point $x_{0}$ of $P(x)$ and $m$ interacting spikes for $v$ near the local maximum point $\bar{x}_{0}$ of $Q(x)$ respectively when $x_{0}\neq \bar{x}_{0}$, moreover, spikes of $u$ and $v$ repel each other. Meanwhile, we prove the attractive phenomenon for $\beta < 0$ and the repulsive phenomenon for $\beta > 0$.
2016, 36(4): 2229-2256 doi: 10.3934/dcds.2016.36.2229 +[Abstract](840) +[PDF](563.1KB)
Abstract:
In this paper, motivated by [13], we use the Littlewood-Paley theory to investigate the Cauchy problem of the Boltzmann equation. When the initial data is a small perturbation of an equilibrium state, under the Grad's angular cutoff assumption, we obtain the unique global strong solution to the Boltzmann equation for the hard potential case in the Chemin-Lerner type spaces $C([0,\infty);\widetilde{L}^{2}_{\xi}(B_{2,r}^{s}))$ with $1\leq r\leq2$ and $s>3/2$ or $s=3/2$ and $r=1$. Besides, we also prove the Lipschitz continuity of the solution map. Our results extend some previous works on the Boltzmann equation in Sobolev spaces.
2016, 36(4): 2257-2284 doi: 10.3934/dcds.2016.36.2257 +[Abstract](744) +[PDF](713.8KB)
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We investigate existence, nonexistence and uniqueness of positive solutions of critical Schrödinger-Poisson systems in closed $4$-manifolds. In the process we provide a sharp criterion for the non-existence of resonant states.
2016, 36(4): 2285-2303 doi: 10.3934/dcds.2016.36.2285 +[Abstract](858) +[PDF](439.8KB)
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In earlier work, Lomeli and Meiss [9] used a generalization of the symplectic approach to study volume preserving generating differential forms. In particular, for the $\mathbb{R}^3$ case, the first to differ from the symplectic case, they derived thirty-six one-forms that generate exact volume preserving maps. In [20], Xue and Zanna studied these differential forms in connection with the numerical solution of divergence-free differential equations: can such forms be used to devise new volume preserving integrators or to further understand existing ones? As a partial answer to this question, Xue and Zanna showed how six of the generating volume form were naturally associated to consistent, first order, volume preserving numerical integrators. In this paper, we investigate and classify the remaining cases. The main result is the reduction of the thirty-six cases to five essentially different cases, up to variable relabeling and adjunction. We classify these five cases, identifying two novel classes and associating the other three to volume preserving vector fields under a Hamiltonian or Lagrangian representation. We demonstrate how these generating form lead to consistent volume preserving schemes for volume preserving vector fields in $\mathbb{R}^3$.
2016, 36(4): 2305-2328 doi: 10.3934/dcds.2016.36.2305 +[Abstract](918) +[PDF](518.0KB)
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In this paper, we are concerned with superlinear impact oscillators of Hill's type with indefinite weight $$\left\{\begin{array}{lll} x''+f(x)x'+q(t)g(x)=0, ~\text{for}~ x(t)>0;\\ x(t)\geq0;\\ x'(t_0+)=-x'(t_0-),~\text{if}~x(t_0)=0,\end{array}\right.$$ where the indefinite weight $q(t)$, defined in $(a,b)$ with $-\infty\leq a< b \leq+\infty,$ has infinitely many zeros in $(a,b),$ $g$ is superlinear and $f$ is bounded. We prove the existence of globally defined bouncing solutions with prescribed number of impacts in the intervals of negativity and positivity of $q$. Furthermore, we show that when $q$ is periodic, the equation under consideration exhibits an interesting phenomenon of chaotic-like dynamics. Finally, in case that $q$ is even and periodic, we prove the existence and multiplicity of the even and periodic bouncing solutions for the Hill's type equation in case of $f\equiv0.$
2016, 36(4): 2329-2346 doi: 10.3934/dcds.2016.36.2329 +[Abstract](997) +[PDF](418.1KB)
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We are interested in finding entire solutions of a bistable periodic lattice dynamical system. By constructing appropriate super- and subsolutions of the system, we establish two different types of merging-front entire solutions. The first type can be characterized by two monostable fronts merging and converging to a single bistable front; while the second type is a solution which behaves as a monostable front merging with a bistable front and one chases another from the same side of $x$-axis. For this discrete and spatially periodic system, we have to emphasize that there has no symmetry between the increasing and decreasing pulsating traveling fronts, which increases the difficulty of construction of the super- and subsolutions.
2016, 36(4): 2347-2364 doi: 10.3934/dcds.2016.36.2347 +[Abstract](956) +[PDF](566.6KB)
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We derive conditions on the initial data, including cases where the initial momentum density is not of one sign, that produce blow-up of the induced solution to the periodic modified Camassa-Holm equation with cubic nonlinearity. The blow-up conditions and the blow-up rate are formulated in terms of the initial momentum density and the average initial energy.
2016, 36(4): 2365-2366 doi: 10.3934/dcds.2016.36.2365 +[Abstract](662) +[PDF](183.8KB)
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