# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

October 2017 , Volume 37 , Issue 10

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2017, 37(10): 5065-5083 doi: 10.3934/dcds.2017219 +[Abstract](932) +[HTML](28) +[PDF](426.6KB)
Abstract:

In this paper, we study the global existence and stability problem of a perturbed viscous circulatory flow around a disc. This flow is described by two-dimensional Navier-Stokes equations. By introducing some suitable weighted energy space and establishing a priori estimates, we show that the 2-D circulatory flow is globally stable in time when the corresponding initial-boundary values are perturbed sufficiently small.

2017, 37(10): 5085-5104 doi: 10.3934/dcds.2017220 +[Abstract](774) +[HTML](23) +[PDF](1167.8KB)
Abstract:

We prove that any \begin{document}$C^{1+\text{BV}}$\end{document} degree d ≥ 2 circle covering \begin{document}$h$\end{document} having all periodic orbits weakly expanding, is conjugate by a \begin{document}$C^{1+\text{BV}}$\end{document} diffeomorphism to a metrically expanding map. We use this to connect the space of parabolic external maps (coming from the theory of parabolic-like maps) to metrically expanding circle coverings.

2017, 37(10): 5105-5125 doi: 10.3934/dcds.2017221 +[Abstract](1492) +[HTML](26) +[PDF](446.1KB)
Abstract:

In this paper, we study the explosive solutions to a class of parbolic stochastic semilinear differential equations driven by a Lévy type noise. The sufficient conditions are presented to guarantee the existence of a unique positive solution of the stochastic partial differential equation under investigation. Moreover, we show that positive solutions will blow up in finite time in mean Lp-norm sense, provided that the initial data, the nonlinear term and the multiplicative noise satisfies some conditions. Several examples are presented to illustrate the theory. Finally, we establish a global existence theorem based on a Lyapunov functional and prove that a stochastic Allen-Cahn equation driven by Lévy noise has a global solution.

2017, 37(10): 5127-5149 doi: 10.3934/dcds.2017222 +[Abstract](828) +[HTML](34) +[PDF](818.1KB)
Abstract:

We study the global bifurcation and exact multiplicity of positive solutions for the positone multiparameter problem

where λ > 0 is a bifurcation parameter and \begin{document}$\varepsilon >0$\end{document} is an evolution parameter. Under some suitable hypotheses on \begin{document}$f_{\varepsilon }(u)$\end{document}, we prove that there exists \begin{document}$\tilde{\varepsilon}>0$\end{document} such that, on the \begin{document}$(λ ,||u||_{∞ })$\end{document}-plane, the bifurcation curve is S-shaped for \begin{document}$0<\varepsilon <\tilde{\varepsilon}$\end{document} and is monotone increasing for \begin{document}$\varepsilon ≥ \tilde{\varepsilon}$\end{document}. We give an application for this problem with a class of polynomial nonlinearities \begin{document}$f_{\varepsilon}(u)=-\varepsilon u^{p}+bu^{2}+cu+d\$\end{document} of degree p≥ 3 and coefficients \begin{document}$\varepsilon ,b,d>0,$\end{document} c ≥ 0. Our results generalize those in Hung and Wang (Trans. Amer. Math. Soc. 365 (2013) 1933-1956.)

2017, 37(10): 5151-5162 doi: 10.3934/dcds.2017223 +[Abstract](864) +[HTML](26) +[PDF](367.8KB)
Abstract:

We study the long-time asymptotic behaviour of viscosity solutions $u(x,~t)$ of the Hamilton-Jacobi equation \begin{document}$u_t(x, t)+ H(x, u(x, t),$\end{document} \begin{document}$Du(x, t))= 0$\end{document} in \begin{document}$\mathbb{T}^n× {(-∞, ∞)}$\end{document}, where \begin{document}$H= H(x, u, p)$\end{document} is convex and coercive in p and non-decreasing on u, and establish the uniform convergence of u to an an asymptotic solution u as $t~\to \text{ }\infty$. Moreover, u is a viscosity solution of Hamilton-Jacobi equation \begin{document}$H(x, u(x), Du(x))= 0$\end{document}.

2017, 37(10): 5163-5190 doi: 10.3934/dcds.2017224 +[Abstract](937) +[HTML](69) +[PDF](524.9KB)
Abstract:

The aim of the paper is to unify the efforts in the study of integrable billiards within quadrics in flat and curved spaces and to explore further the interplay of symplectic and contact integrability. As a starting point in this direction, we consider virtual billiard dynamics within quadrics in pseudo-Euclidean spaces. In contrast to the usual billiards, the incoming velocity and the velocity after the billiard reflection can be at opposite sides of the tangent plane at the reflection point. In the symmetric case we prove noncommutative integrability of the system and give a geometrical interpretation of integrals, an analog of the classical Chasles and Poncelet theorems and we show that the virtual billiard dynamics provides a natural framework in the study of billiards within quadrics in projective spaces, in particular of billiards within ellipsoids on the sphere \begin{document}${\mathbb{S}^{n - 1}}$\end{document} and the Lobachevsky space \begin{document}$\mathbb H^{n-1}$\end{document}.

2017, 37(10): 5191-5209 doi: 10.3934/dcds.2017225 +[Abstract](1068) +[HTML](15) +[PDF](501.2KB)
Abstract:

We extend the Phase Transition model for traffic proposed in [8], by Colombo, Marcellini, and Rascle to the network case. More precisely, we consider the Riemann problem for such a system at a general junction with \begin{document}$n$\end{document} incoming and \begin{document}$m$\end{document} outgoing roads. We propose a Riemann solver at the junction which conserves both the number of cars and the maximal speed of each vehicle, which is a key feature of the Phase Transition model. For special junctions, we prove that the Riemann solver is well defined.

2017, 37(10): 5211-5252 doi: 10.3934/dcds.2017226 +[Abstract](645) +[HTML](44) +[PDF](660.4KB)
Abstract:

We study the behavior as \begin{document}$t \to 0^+$\end{document} of nonnegative functions

satisfying the parabolic Choquard-Pekar type inequalities

where \begin{document}$α∈(0, n+2)$\end{document}, \begin{document}$λ>0$\end{document}, and \begin{document}$σ≥0$\end{document} are constants, \begin{document}$Φ$\end{document} is the heat kernel, and \begin{document}$*$\end{document} is the convolution operation in \begin{document}$\mathbb{R}^n× (0, 1)$\end{document}. We provide optimal conditions on \begin{document}$α, λ$\end{document}, and \begin{document}$σ$\end{document} such that nonnegative solutions \begin{document}$u$\end{document} of (0.1), (0.2) satisfy pointwise bounds in compact subsets of \begin{document}$B_1(0)$\end{document} as \begin{document}$t \to0^+$\end{document}. We obtain similar results for nonnegative solutions of (0.1), (0.2) when \begin{document}$Φ^{α/n}$\end{document} in (0.2) is replaced with the fundamental solution \begin{document}$Φ_α$\end{document} of the fractional heat operator \begin{document}$(\frac{\partial}{\partial t}-Δ)^{α/2}$\end{document}.

2017, 37(10): 5253-5269 doi: 10.3934/dcds.2017227 +[Abstract](702) +[HTML](22) +[PDF](206.1KB)
Abstract:

We exhibit different examples of minimal sets for an IFS of homeomorphisms with rational rotation number. It is proved that these examples are, from a topological point of view, the unique possible cases.

2017, 37(10): 5271-5284 doi: 10.3934/dcds.2017228 +[Abstract](719) +[HTML](21) +[PDF](419.1KB)
Abstract:

We consider a multi-dimensional billiard system in an \begin{document}$(n+1)$\end{document}-dimensional Euclidean space, the direct product of the "horizontal" hyperplane and the "vertical" line. The hypersurface that determines the system is assumed to be smooth and symmetric in all coordinate hyperplanes. Hence there exists a periodic orbit \begin{document}$γ$\end{document} of period 2 moving along the "vertical" coordinate axis. The question we ask is as follows. Is it possible to choose such a system to have the dynamics locally (near \begin{document}$γ$\end{document}) conjugated to the dynamics of a linear map?

Since the problem is local, the billiard hypersurface can be determined as the graphs of the functions \begin{document}$± f$\end{document}, where \begin{document}$f$\end{document} is even and defined in a neighborhood of the origin on the "horizontal" coordinate hyperplane. We prove that \begin{document}$f$\end{document} exists as a formal Taylor series in the non-resonant case and give numerical evidence for convergence of the series.

2017, 37(10): 5285-5297 doi: 10.3934/dcds.2017229 +[Abstract](736) +[HTML](20) +[PDF](379.6KB)
Abstract:

In this short note we establish a law of large permanent for matrices with entries from an \begin{document}$\mathbf{N}^2$\end{document}-indexed stochastic process. This answers a question by Bochi, Iommi and Ponce in [4].

2017, 37(10): 5299-5317 doi: 10.3934/dcds.2017230 +[Abstract](789) +[HTML](21) +[PDF](440.0KB)
Abstract:

In this paper, we consider the following semilinear elliptic problem

where $\frac{N}{N-2} < q < p < p^*$ or $q>p>p^*$, $p^*=\frac{N+2}{N-2}$, $N≥3$. We show that if $q$ is fixed and $p$ is close enough to $\frac{N+2}{N-2}$, the above problem has radial nodal bubble tower solutions, which behave like a superposition of bubbles with different orders and blow up at the origin.

2017, 37(10): 5319-5335 doi: 10.3934/dcds.2017231 +[Abstract](921) +[HTML](19) +[PDF](433.3KB)
Abstract:

In the paper we study the problem of the influence of the parametric uncertainties on the Bohl exponents of discrete time-varying linear system. We obtain formulas for the computation of the exact boundaries of lower and upper mobility for the supremum and infimum of the Bohl exponents under arbitrary small perturbations of system coefficients matrices on the basis of the transition matrix.

2017, 37(10): 5337-5354 doi: 10.3934/dcds.2017232 +[Abstract](958) +[HTML](24) +[PDF](422.0KB)
Abstract:

We study the Bonsall cone spectral radius and the approximate point spectrum of (in general non-linear) positively homogeneous, bounded and supremum preserving maps, defined on a max-cone in a given normed vector lattice. We prove that the Bonsall cone spectral radius of such maps is always included in its approximate point spectrum. Moreover, the approximate point spectrum always contains a (possibly trivial) interval. Our results apply to a large class of (nonlinear) max-type operators.

We also generalize a known result that the spectral radius of a positive (linear) operator on a Banach lattice is contained in the approximate point spectrum. Under additional generalized compactness type assumptions our results imply Krein-Rutman type results.

2017, 37(10): 5355-5365 doi: 10.3934/dcds.2017233 +[Abstract](845) +[HTML](32) +[PDF](383.8KB)
Abstract:

In this paper, we prove Strichartz estimates for \begin{document}$N$\end{document}-body Schrödinger operators, provided that interaction potentials are small enough. Our tools are new Strichartz estimates with frozen spatial variables, and its improvement in the \begin{document}$V_S^p$\end{document}-norm of Koch and Tataru [19]. As an application, we prove scattering for \begin{document}$N$\end{document}-body Schrödinger operators.

2017, 37(10): 5367-5405 doi: 10.3934/dcds.2017234 +[Abstract](1006) +[HTML](18) +[PDF](633.6KB)
Abstract:

We describe the multifractal nature of random weak Gibbs measures on some classes of attractors associated with \begin{document}$C^1$\end{document} random dynamics semi-conjugate to a random subshift of finite type. This includes the validity of the multifractal formalism, the calculation of Hausdorff and packing dimensions of the so-called level sets of divergent points, and a \begin{document}$0$\end{document}-\begin{document}$∞$\end{document} law for the Hausdorff and packing measures of the level sets of the local dimension.

2017, 37(10): 5407-5431 doi: 10.3934/dcds.2017235 +[Abstract](1449) +[HTML](25) +[PDF](497.4KB)
Abstract:

In this article we obtain a variational principle for saturated sets for maps with some non-uniform specification properties. More precisely, we prove that the topological entropy of saturated sets coincides with the smallest measure theoretical entropy among the invariant measures in the accumulation set. Using this fact we provide lower bounds for the topological entropy of the irregular set and the level sets in the multifractal analysis of Birkhoff averages for continuous observables. The topological entropy estimates use as tool a non-uniform specification property on topologically large sets, which we prove to hold for open classes of non-uniformly expanding maps. In particular we prove some multifractal analysis results for C1-open classes of non-uniformly expanding local diffeomorphisms and Viana maps [1,33].

2017, 37(10): 5433-5454 doi: 10.3934/dcds.2017236 +[Abstract](1162) +[HTML](37) +[PDF](455.5KB)
Abstract:

This paper will mainly study the information about the existence and stability of the invasion traveling waves for the nonlocal Leslie-Gower predator-prey model. By using an invariant cone in a bounded domain with initial function being defined on and applying the Schauder's fixed point theorem, we can obtain the existence of traveling waves. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Then we use the weighted energy to prove that the invasion traveling waves are exponentially stable as perturbation in some exponentially as \begin{document}$x\to-\infty$\end{document}. Finally, by defining the bilateral Laplace transform, we can obtain the nonexistence of the traveling waves.

2017, 37(10): 5455-5465 doi: 10.3934/dcds.2017237 +[Abstract](563) +[HTML](19) +[PDF](314.7KB)
Abstract:

We show that for any set of \begin{document} $n$ \end{document} distinct points in the complex plane, there exists a polynomial \begin{document} $p$ \end{document} of degree at most \begin{document} $n+1$ \end{document} so that the corresponding Halley and Schröder map for \begin{document} $p$ \end{document} has the given points as a super-attracting cycle. This improves the result in [1], which shows how to find such a polynomial of degree \begin{document} $3n$ \end{document}. Moreover we show that in general one cannot improve upon degree \begin{document} $n+1$ \end{document}.

2017, 37(10): 5467-5502 doi: 10.3934/dcds.2017238 +[Abstract](820) +[HTML](30) +[PDF](602.5KB)
Abstract:

Let Ω be a bounded domain in \begin{document}$\mathbb{R}^2$\end{document} with smooth boundary, we study the following Neumann boundary value problem

where \begin{document} $ν$ \end{document} denotes the outer unit normal vector to \begin{document} $\partial \Omega$ \end{document}, \begin{document} $h∈ C^{0,α}(\partial \Omega)$ \end{document}, \begin{document} $s>0$ \end{document} is a large parameter and \begin{document} $\phi_1$ \end{document} is a positive first Steklov eigenfunction. We construct solutions of this problem which exhibit multiple boundary concentration behavior around maximum points of \begin{document} $\phi_1$ \end{document} on the boundary as \begin{document} $s\to+∞$ \end{document}.

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