# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

December 2017 , Volume 37 , Issue 12

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2017, 37(12): 5979-6034 doi: 10.3934/dcds.2017259 +[Abstract](2468) +[HTML](121) +[PDF](801.1KB)
Abstract:

We consider a system describing the long-time dynamics of an hydrodynamical, density-dependent flow under the effects of gravitational forces. We prove that if the Froude number is sufficiently small such system is globally well posed with respect to a \begin{document}$H^s, \ s>1/2$\end{document} Sobolev regularity. Moreover if the Froude number converges to zero we prove that the solutions of the aforementioned system converge (strongly) to a stratified three-dimensional Navier-Stokes system. No smallness assumption is assumed on the initial data.

2017, 37(12): 6035-6067 doi: 10.3934/dcds.2017260 +[Abstract](3149) +[HTML](76) +[PDF](616.7KB)
Abstract:

In this paper, we study the dynamics of a non-autonomous reaction diffusion model with the fractional diffusion on the whole space. We firstly prove the existence of a \begin{document}$(L^2,L^2)$\end{document} pullback \begin{document}$\mathscr{D}_μ$\end{document} -attractor of this model. Then we show that the pullback \begin{document}$\mathscr{D}_μ$\end{document} -attractor attract the \begin{document}$\mathscr{D}_μ$\end{document} class (especially all \begin{document}$L^2$\end{document} -bounded set) in $L^{2+δ}$-norm for any \begin{document}$δ∈[0,∞)$\end{document}. Moreover, the solution of the model is shown to be continuous in \begin{document}$H^s$\end{document} with respect to initial data under a slightly stronger condition on external forcing term. As an application, we prove that the \begin{document}$(L^2,L^2)$\end{document} pullback $\mathscr{D}_{μ}$-attractor indeed attract the class of \begin{document}$\mathscr{D}_{μ}$\end{document} in \begin{document}$H^s$\end{document} -norm, and thus the existence of a \begin{document}$(L^2, H^s)$\end{document} pullback \begin{document}$\mathscr{D}_μ$\end{document} -attractor is obtained.

2017, 37(12): 6069-6098 doi: 10.3934/dcds.2017261 +[Abstract](2109) +[HTML](71) +[PDF](664.5KB)
Abstract:

We introduce a family of kinetic vector fields on countable space-time grids and study related impulsive second order initial value Cauchy problems. We then construct special examples for which orbits and attractors display unusual analytic and geometric properties.

2017, 37(12): 6099-6121 doi: 10.3934/dcds.2017262 +[Abstract](3623) +[HTML](77) +[PDF](559.1KB)
Abstract:

This paper deals with the homogeneous Neumann boundary-value problem for the chemotaxis-consumption system

in \begin{document}$N$\end{document}-dimensional bounded smooth domains for suitably regular positive initial data.

We shall establish the existence of a global bounded classical solution for suitably large $μ$ and prove that for any \begin{document}$μ>0$\end{document} there exists a weak solution.

Moreover, in the case of \begin{document}$κ>0$\end{document} convergence to the constant equilibrium \begin{document}$(\frac{κ}{μ },0)$\end{document} is shown.

2017, 37(12): 6123-6138 doi: 10.3934/dcds.2017263 +[Abstract](3137) +[HTML](76) +[PDF](567.5KB)
Abstract:

It is extremely difficult to establish the existence of almost periodic solutions for delay differential equations via methods that need the compactness conditions such as Schauder's fixed point theorem. To overcome this difficulty, in this paper, we employ a novel technique to construct a contraction mapping, which enables us to establish the existence of almost periodic solution for a delay differential equation system with time-varying coefficients. When the system's coefficients are periodic, coincide degree theory is used to establish the existence of periodic solutions. Global stability results are also obtained by the method of Liapunov functionals.

2017, 37(12): 6139-6152 doi: 10.3934/dcds.2017264 +[Abstract](2557) +[HTML](74) +[PDF](421.8KB)
Abstract:

In this paper, we describe several different meanings for the concept of Gibbs measure on the lattice \begin{document}$\mathbb{N}$\end{document} in the context of finite alphabets (or state space). We compare and analyze these ''in principle" distinct notions: DLR-Gibbs measures, Thermodynamic Limit and eigenprobabilities for the dual of the Ruelle operator (also called conformal measures).

Among other things we extended the classical notion of a Gibbsian specification on \begin{document}$\mathbb{N}$\end{document} in such way that the similarity of many results in Statistical Mechanics and Dynamical System becomes apparent. One of our main result claims that the construction of the conformal Measures in Dynamical Systems for Walters potentials, using the Ruelle operator, can be formulated in terms of Specification. We also describe the Ising model, with \begin{document}$1/r^{2+\varepsilon}$\end{document} interaction energy, in the Thermodynamic Formalism setting and prove that its associated potential is in Walters space -we present an explicit expression. We also provide an alternative way for obtaining the uniqueness of the DLR-Gibbs measures.

2017, 37(12): 6153-6164 doi: 10.3934/dcds.2017265 +[Abstract](1976) +[HTML](78) +[PDF](377.1KB)
Abstract:

In this paper, we continue to study the Calabi flow on complex tori. We develop a new method to obtain an explicit bound of the curvature of the Calabi flow. As an application, we show that when $n=2$, the Calabi flow starting from a weak Kähler metric will become smooth immediately. It implies that in our settings, the weak minimizer of the Mabuchi energy is a smooth one.

2017, 37(12): 6165-6181 doi: 10.3934/dcds.2017266 +[Abstract](2497) +[HTML](67) +[PDF](512.1KB)
Abstract:

For \begin{document} $\mathrm{H}∈ C^2(\mathbb{R}^{N\times n})$ \end{document} and \begin{document} $u :Ω \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$ \end{document}, consider the system

\begin{document}$\label{1}\mathrm{A}_∞ u\, :=\,\Big(\mathrm{H}_P \otimes \mathrm{H}_P + \mathrm{H}[\mathrm{H}_P]^\bot \mathrm{H}_{PP}\Big)(\text{D} u):\text{D}^2u\, =\,0. \tag{1}$ \end{document}

We construct \begin{document} $\mathcal{D}$ \end{document}-solutions to the Dirichlet problem for (1), an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our \begin{document} $\mathcal{D}$ \end{document}-solutions are \begin{document} $W^{1,∞}$ \end{document}-submersions and are obtained without any convexity hypotheses for \begin{document} $\mathrm{H}$ \end{document}, through a result of independent interest involving existence of strong solutions to the singular value problem for general dimensions \begin{document} $n≠ N$ \end{document}.

2017, 37(12): 6183-6188 doi: 10.3934/dcds.2017267 +[Abstract](2086) +[HTML](72) +[PDF](315.0KB)
Abstract:

The Douady's formula was defined for the external argument on the boundary points of the main hyperbolic component \begin{document} $W_0$ \end{document} of the Mandelbrot set \begin{document} $M$ \end{document} and it is given by the map \begin{document} $T(θ)=1/2+θ/4$ \end{document}. We extend this formula to the boundary of all hyperbolic components of \begin{document} $M$ \end{document} and we give a characterization of the parameter in \begin{document} $M$ \end{document} with these external arguments.

2017, 37(12): 6189-6225 doi: 10.3934/dcds.2017268 +[Abstract](3115) +[HTML](81) +[PDF](654.6KB)
Abstract:

The current paper is devoted to the study of spreading speeds and traveling wave solutions of the following parabolic-elliptic chemotaxis system,

where $u(x, t)$ represents the population density of a mobile species and $v(x, t)$ represents the population density of a chemoattractant, and $χ$ represents the chemotaxis sensitivity. We first give a detailed study in the case $N=1$. In this case, it has been shown in an earlier work by the authors of the current paper that, when $0 < χ < 1$, for every nonnegative uniformly continuous and bounded function $u_0(x)$, the system has a unique globally bounded classical solution $(u(x, t;u_0), v(x, t;u_0))$ with initial condition $u(x, 0;u_0)=u_0(x)$. Furthermore, it was shown that, if $0 < χ < \frac{1}{2}$, then the constant steady-state solution $(1, 1)$ is asymptotically stable with respect to strictly positive perturbations. In the current paper, we show that if $0 < χ < 1$, then there are nonnegative constants $c_ - ^*\left( \chi \right) \le c_ + ^*\left( \chi \right)$ such that for every nonnegative initial function $u_0(·)$ with non-empty and compact support ${\rm{supp}}(u_0)$,

and

We also show that if $0 < χ < \frac{1}{2}$, there is a positive constant $c^*(χ)$ such that for every $c \ge c^*(χ)$, the system has a traveling wave solution $(u(x, t), v(x, t))$ with speed $c$ and connecting $(1, 1)$ and $(0, 0)$, that is, $(u(x, t), v(x, t))=(U(x-ct), V(x-ct))$ for some functions $U(·)$ and $V(·)$ satisfying $(U(-∞), V(-∞))=(1, 1)$ and $(U(∞), V(∞))=(0, 0)$. Moreover, we show that

We then consider the extensions of the results in the case $N=1$ to the case $N \ge 2$.

2017, 37(12): 6227-6242 doi: 10.3934/dcds.2017269 +[Abstract](2454) +[HTML](77) +[PDF](411.3KB)
Abstract:

In this paper, we prove the stability of half-degree point defect profiles in \begin{document} $\mathbb{R}^2$ \end{document} for the nematic liquid crystal within Landau-de Gennes model.

2017, 37(12): 6243-6255 doi: 10.3934/dcds.2017270 +[Abstract](2442) +[HTML](71) +[PDF](382.0KB)
Abstract:

Let \begin{document} $Ω\subset\mathbb{R}^N$ \end{document} (\begin{document} $N≥2$ \end{document}) be a bounded domain with Lipschitz boundary. For each \begin{document} $p∈(1,∞)$ \end{document} and \begin{document} $s∈ (0,1)$ \end{document} we denote by \begin{document} $(-Δ_p)^s$ \end{document} the fractional \begin{document} $(s,p)$ \end{document}-Laplacian operator. In this paper we study the existence of nontrivial solutions for a perturbation of the eigenvalue problem \begin{document} $(-Δ_p)^s u=λ |u|^{p-2}u$ \end{document}, in \begin{document} $Ω$ \end{document}, \begin{document} $u=0$ \end{document}, in \begin{document} $\mathbb{R}^N\backslash Ω$ \end{document}, with a fractional \begin{document} $(t,q)$ \end{document}-Laplacian operator in the left-hand side of the equation, when \begin{document} $t∈(0,1)$ \end{document} and \begin{document} $q∈(1,∞)$ \end{document} are such that \begin{document} $s-N/p=t-N/q$ \end{document}. We show that nontrivial solutions for the perturbed eigenvalue problem exists if and only if parameter \begin{document} $λ$ \end{document} is strictly larger than the first eigenvalue of the \begin{document} $(s,p)$ \end{document}-Laplacian.

2017, 37(12): 6257-6289 doi: 10.3934/dcds.2017271 +[Abstract](2508) +[HTML](76) +[PDF](616.7KB)
Abstract:

We analyze rarefaction wave interactions of self-similar transonic irrotational flow in gas dynamics for two dimensional Riemann problems. We establish the existence result of the supersonic solution to the prototype nonlinear wave system for the sectorial Riemann data, and study the formation of the sonic boundary and the transonic shock. The transition from the sonic boundary to the shock boundary inherits at least two types of degeneracies (1) the system is sonic, and in addition (2) the angular derivative of the solution becomes zero where the sonic and shock boundaries meet.

2017, 37(12): 6291-6318 doi: 10.3934/dcds.2017272 +[Abstract](2926) +[HTML](77) +[PDF](526.0KB)
Abstract:

In this paper, we provide a general approach to study the asymptotic behavior of traveling wave solutions for a three-component system with nonlocal dispersal. Then as an important application, we establish a new type of entire solutions which behave as two traveling wave solutions coming from both sides of \begin{document} $x$ \end{document}-axis for a three-species Lotka-Volterra competition system.

2017, 37(12): 6319-6331 doi: 10.3934/dcds.2017273 +[Abstract](2516) +[HTML](61) +[PDF](355.4KB)
Abstract:

By considering all possible reparametrizations of the flows instead of the time-\begin{document} $1$ \end{document} maps, we introduce Bowen topological entropy and local entropy on subsets for flows. Through handling techniques for reparametrization balls, we prove a covering lemma for fixed-point free flows and then prove a variational principle.

2017, 37(12): 6333-6352 doi: 10.3934/dcds.2017274 +[Abstract](1885) +[HTML](60) +[PDF](481.0KB)
Abstract:

We investigate the well-posedness, the exponential stability, or the lack thereof, of thermoelastic systems in materials where, in contrast to classical thermoelastic models for Kirchhoff type plates, two temperatures are involved, related by an elliptic equation. The arising initial boundary value problems for different boundary conditions deal with systems of partial differential equations involving Schrödinger like equations, hyperbolic and elliptic equations, which have a different character compared to the classical one with the usual single temperature. Depending on the model -with Fourier or with Cattaneo type heat conduction -we obtain exponential resp. non-exponential stability, thus providing another examples where the change from Fourier's to Cattaneo's law leads to a loss of exponential stability.

2017, 37(12): 6353-6368 doi: 10.3934/dcds.2017275 +[Abstract](2829) +[HTML](78) +[PDF](376.2KB)
Abstract:

We prove that, for semi-invertible linear cocycles, Lyapunov exponents of ergodic measures may be approximated by Lyapunov exponents on periodic points.

2017, 37(12): 6369-6382 doi: 10.3934/dcds.2017276 +[Abstract](2267) +[HTML](67) +[PDF](376.2KB)
Abstract:

In this paper we will give the optimal upper bound for the first eigenvalue of the fourth order equation with integrable potentials when the L1 norm of potentials is known. Combining with the results for the corresponding optimal lower bound problem in [12], we have completely obtained the optimal estimation for the first eigenvalue of the fourth order equation.

2017, 37(12): 6383-6403 doi: 10.3934/dcds.2017277 +[Abstract](2583) +[HTML](69) +[PDF](482.8KB)
Abstract:

We derive general existence theorems for random pullback exponential attractors and deduce explicit bounds for their fractal dimension. The results are formulated for asymptotically compact random dynamical systems in Banach spaces.

2017, 37(12): 6405-6435 doi: 10.3934/dcds.2017278 +[Abstract](1836) +[HTML](80) +[PDF](600.1KB)
Abstract:

A multi-dimensional junction is obtained by identifying the boundaries of a finite number of copies of an Euclidian half-space. The main contribution of this article is the construction of a multidimensional vertex test function G(x, y). First, such a function has to be sufficiently regular to be used as a test function in the viscosity solution theory for quasi-convex Hamilton-Jacobi equations posed on a multi-dimensional junction. Second, its gradients have to satisfy appropriate compatibility conditions in order to replace the usual quadratic penalization function |x-y|2 in the proof of strong uniqueness (comparison principle) by the celebrated doubling variable technique. This result extends a construction the authors previously achieved in the network setting. In the multi-dimensional setting, the construction is less explicit and more delicate.

2017, 37(12): 6437-6470 doi: 10.3934/dcds.2017279 +[Abstract](2238) +[HTML](58) +[PDF](376.2KB)
Abstract:

This paper is dedicated to the global well-posedness issue of the compressible Oldroyd-B model in the whole space \begin{document}$\mathbb{R}^d$\end{document} with \begin{document}$d≥2$\end{document}. By exploiting the intrinsic structure of the system, we prove that if the initial data is small enough (depending on the coupling parameter), this set of equations admits a unique global solution in a certain critical Besov space. This result partially improves the previous work by Fang and the author [J. Differential Equations, 256(2014), 2559-2602].

2017, 37(12): 6471-6485 doi: 10.3934/dcds.2017280 +[Abstract](3064) +[HTML](80) +[PDF](423.9KB)
Abstract:

In this paper, we mainly study the Cauchy problem of an integrable dispersive Hunter-Saxton equation in periodic domain. Firstly, we establish local well-posedness of the Cauchy problem of the equation in \begin{document}$H^s (\mathbb{S}), s > \frac{3}{2},$\end{document} by applying the Kato method. Secondly, by using some conservative quantities, we give a precise blow-up criterion and a blow-up result of strong solutions to the equation. Finally, based on a sign-preserve property, we transform the original equation into the sinh-Gordon equation. By using the travelling wave solutions of the sinh-Gordon equation and a period stretch between these two equations, we get the travelling wave solutions of the original equation.

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