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Discrete & Continuous Dynamical Systems - A

2018 , Volume 38 , Issue 5

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Decaying turbulence for the fractional subcritical Burgers equation
Alexandre Boritchev
2018, 38(5): 2229-2249 doi: 10.3934/dcds.2018092 +[Abstract](45) +[HTML](19) +[PDF](490.54KB)

We consider the fractional unforced Burgers equation in the one-dimensional space-periodic setting:

Here \begin{document}$ f$\end{document} is strongly convex and satisfies a growth condition, \begin{document}$ Λ = \sqrt{-Δ}, \ ν$\end{document} is small and positive, while \begin{document}$ α ∈ (1,\ 2)$\end{document} is a constant in the subcritical range.

For solutions \begin{document}$ u$\end{document} of this equation, we generalise the results obtained for the case \begin{document}$ α = 2$\end{document} (i.e. when \begin{document}$ -Λ^{α}$\end{document} is the Laplacian) in [12]. We obtain sharp estimates for the time-averaged Sobolev norms of \begin{document}$ u$\end{document} as a function of \begin{document}$ ν$\end{document}. These results yield sharp \begin{document}$ν$\end{document}-independent estimates for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. In the inertial range, these quantities behave as a power of the norm of the relevant parameter, which is respectively the separation \begin{document}$ \ell$\end{document} in the physical space and the wavenumber \begin{document}$ \bf{k}$\end{document} in the Fourier space.

The form of all estimates is the same as in the case \begin{document}$ α = 2$\end{document}; the only thing which changes is that \begin{document}$ ν$\end{document} is replaced by \begin{document}$ ν^{1/(α-1)}$\end{document}.

Global stability of V-shaped traveling fronts in combustion and degenerate monostable equations
Zhen-Hui Bu and Zhi-Cheng Wang
2018, 38(5): 2251-2286 doi: 10.3934/dcds.2018093 +[Abstract](37) +[HTML](16) +[PDF](627.88KB)

This paper is concerned with the global stability of V-shaped traveling fronts in reaction-diffusion equations with combustion and degenerate monostable nonlinearity. The existence of such curved fronts has been recently proved by [39]. In this paper, by constructing new subsolutions, we show the asymptotic stability of V-shaped traveling fronts.

Dichotomy spectrum and almost topological conjugacy on nonautonomus unbounded difference systems
Álvaro Castañeda and Gonzalo Robledo
2018, 38(5): 2287-2304 doi: 10.3934/dcds.2018094 +[Abstract](31) +[HTML](14) +[PDF](470.31KB)

We construct a bijection between the solutions of a linear system of nonautonomous difference equations which is uniformly asymptotically stable and its unbounded perturbation. The key idea used to made this bijection is to consider the crossing times of the solutions with the unit sphere. This approach prompt us to introduce the concept of almost topological conjugacy in this nonautonomous framework. This task is carried out by simplifying both systems through a spectral approach of the notion of almost reducibility combined with suitable technical assumptions.

KdV-like solitary waves in two-dimensional FPU-lattices
Fanzhi Chen and Michael Herrmann
2018, 38(5): 2305-2332 doi: 10.3934/dcds.2018095 +[Abstract](35) +[HTML](17) +[PDF](1315.59KB)

We prove the existence of solitary waves in the KdV limit of two-dimensional FPU-type lattices using asymptotic analysis of nonlinear and singularly perturbed integral equations. In particular, we generalize the existing results by Friesecke and Matthies since we allow for arbitrary propagation directions and non-unidirectional wave profiles.

Improved results for Klein-Gordon-Maxwell systems with general nonlinearity
Sitong Chen and Xianhua Tang
2018, 38(5): 2333-2348 doi: 10.3934/dcds.2018096 +[Abstract](46) +[HTML](17) +[PDF](445.42KB)

This paper is concerned with the following Klein-Gordon-Maxwell system

where \begin{document}$0 < ω≤ m_0$\end{document} and \begin{document}$f∈ \mathcal{C}(\mathbb{R}, \mathbb{R})$\end{document}. By introducing some new tricks, we prove that the above system has 1) a ground state solution in the case when \begin{document}$0 < ω < m_0$\end{document} and \begin{document}$f$\end{document} is superlinear at infinity; 2) a nontrivial solution in the zero mass case, i.e. \begin{document}$ω = m_0$\end{document} and \begin{document}$f$\end{document} is super-quadratic at infinity. These results improve the related ones in the literature.

Conormal derivative problems for stationary Stokes system in Sobolev spaces
Jongkeun Choi, Hongjie Dong and Doyoon Kim
2018, 38(5): 2349-2374 doi: 10.3934/dcds.2018097 +[Abstract](31) +[HTML](17) +[PDF](491.28KB)

We prove the solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with irregular coefficients on bounded Reifenberg flat domains. The coefficients are assumed to be merely measurable in one direction, which may differ depending on the local coordinate systems, and have small mean oscillations in the other directions. In the course of the proof, we use a local version of the Poincaré inequality on Reifenberg flat domains, the proof of which is of independent interest.

Hausdorff dimension of certain sets arising in Engel continued fractions
Lulu Fang and Min Wu
2018, 38(5): 2375-2393 doi: 10.3934/dcds.2018098 +[Abstract](40) +[HTML](15) +[PDF](468.01KB)

In the present paper, we are concerned with the Hausdorff dimension of certain sets arising in Engel continued fractions. In particular, the Hausdorff dimension of sets

are completely determined, where \begin{document}$i.m.$\end{document} means infinitely many, \begin{document}$\{b_n(x)\}_{n ≥ 1}$\end{document} is the sequence of partial quotients of the Engel continued fraction expansion of \begin{document}$x$\end{document} and \begin{document}$\phi$\end{document} is a positive function defined on natural numbers.

On Hausdorff dimension of the set of non-ergodic directions of two-genus double cover of tori
Yan Huang
2018, 38(5): 2395-2409 doi: 10.3934/dcds.2018099 +[Abstract](25) +[HTML](15) +[PDF](387.03KB)

Cheung, Hubert and Masur [Invent. Math., 183(2011), no.2, pp. 337-383] proved that the Hausdorff dimension of the set of nonergodic directions of billiards in a kind of rectangle with barrier is either 0 or \begin{document}$\frac{1}{2}$\end{document}. As an application of their argument, we prove that there exist the third-kind two-genus double covers of tori in which the set of minimal and non-ergodic directions have Hausdorff dimension \begin{document}$\frac{1}{2}$\end{document}.

Positive periodic solutions of the weighted $p$-Laplacian with nonlinear sources
Shanming Ji, Jingxue Yin and Yutian Li
2018, 38(5): 2411-2439 doi: 10.3934/dcds.2018100 +[Abstract](35) +[HTML](20) +[PDF](573.46KB)

In this paper we study the existence of time periodic solutions for the evolutionary weighted \begin{document}$p$\end{document}-Laplacian with a nonlinear periodic source in a bounded domain containing the origin. We show that there is a critical exponent \begin{document}$q_c = q_c(α,β) = \frac{(N+β)p}{N+α-p}-1$\end{document} and a singular exponent \begin{document}$q_s = p-1$\end{document}: there exists a positive periodic solution when \begin{document}$0<q<q_c$\end{document} and \begin{document}$q\ne q_s$\end{document}; while there is no positive periodic solution when \begin{document}$q≥ q_c$\end{document}. The case when \begin{document}$q = q_s$\end{document} is completely different from the remaining case \begin{document}$q\ne q_s$\end{document}, the problem may or may not have solutions depending on the coefficients of the equation.

The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity
Takashi Kajiwara
2018, 38(5): 2441-2465 doi: 10.3934/dcds.2018101 +[Abstract](26) +[HTML](17) +[PDF](467.03KB)

We construct a heteroclinic solution to the FitzHugh-Nagumo type reaction-diffusion system (FHN RD system) with heterogeneity by the sub-supersolution method due to [5]. \begin{document} $σ(d,γ)$ \end{document} is introduced as the Rayleigh quotient corresponding to a linearized eigenvalue problem of the subsolution, where \begin{document} $d$ \end{document} and \begin{document} $γ$ \end{document} are parameters. The key to construct the solution is the uniform estimate for \begin{document} $σ(·,·)$ \end{document} from below. In addition, it enables us to analyze an asymptotic behavior of the solution.

Wiener-Landis criterion for Kolmogorov-type operators
Alessia E. Kogoj, Ermanno Lanconelli and Giulio Tralli
2018, 38(5): 2467-2485 doi: 10.3934/dcds.2018102 +[Abstract](27) +[HTML](16) +[PDF](439.85KB)

We establish a necessary and sufficient condition for a boundary point to be regular for the Dirichlet problem related to a class of Kolmogorov-type equations. Our criterion is inspired by two classical criteria for the heat equation: the Evans-Gariepy's Wiener test, and a criterion by Landis expressed in terms of a series of caloric potentials.

Topological stability and spectral decomposition for homeomorphisms on noncompact spaces
Keonhee Lee, Ngoc-Thach Nguyen and Yinong Yang
2018, 38(5): 2487-2503 doi: 10.3934/dcds.2018103 +[Abstract](41) +[HTML](19) +[PDF](374.08KB)

In this paper, we introduce the notions of expansiveness, shadowing property and topological stability for homeomorphisms on noncompact metric spaces which are dynamical properties and equivalent to the classical definitions in case of compact metric spaces. Then we extend the Walters's stability theorem and Smale's spectral decomposition theorem to homeomorphisms on locally compact metric spaces.

A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane
Shenghao Li, Min Chen and Bing-Yu Zhang
2018, 38(5): 2505-2525 doi: 10.3934/dcds.2018104 +[Abstract](27) +[HTML](17) +[PDF](499.68KB)

The paper is concerned with an initial-boundary-value problem of the sixth order Boussinesq equation posed on a quarter plane with non-homogeneous boundary conditions:

where \begin{document}$β = ± 1$\end{document} . It is shown that the problem is locally well-posed in the space $H^s(\mathbb{R}^+)$ for any 0≤s<\begin{document}$\frac{13}{2}$\end{document} with the initial data \begin{document}$ (\varphi, ψ)$\end{document} in the space

and the naturally compatible boundary data

with optimal regularity.

Stability of the distribution function for piecewise monotonic maps on the interval
Michal Málek and Peter Raith
2018, 38(5): 2527-2539 doi: 10.3934/dcds.2018105 +[Abstract](25) +[HTML](19) +[PDF](369.68KB)

For piecewise monotonic maps the notion of approximating distribution function is introduced. It is shown that for a mixing basic set it coincides with the usual distribution function. Moreover, it is proved that the approximating distribution function is upper semi-continuous under small perturbations of the map.

Constant slope models for finitely generated maps
Samuel Roth
2018, 38(5): 2541-2554 doi: 10.3934/dcds.2018106 +[Abstract](21) +[HTML](15) +[PDF](450.89KB)

We study countably monotone and Markov interval maps. We establish sufficient conditions for uniqueness of a conjugate map of constant slope. We explain how global window perturbation can be used to generate a large class of maps satisfying these conditions.

Global existence in the critical space for the Thirring and Gross-Neveu models coupled with the electromagnetic field
Sigmund Selberg
2018, 38(5): 2555-2569 doi: 10.3934/dcds.2018107 +[Abstract](29) +[HTML](27) +[PDF](414.32KB)

We prove global well-posedness for the coupled Maxwell-Dirac-Thirring-Gross-Neveu equations in one space dimension, with data for the Dirac spinor in the critical space $L^2(\mathbb{R})$. In particular, we recover earlier results of Candy and Huh for the Thirring and Gross-Neveu models, respectively, without the coupling to the electromagnetic field, but the function spaces we introduce allow for a greatly simplified proof. We also apply our method to prove local well-posedness in $L^2(\mathbb{R})$ for a quadratic Dirac equation, improving an earlier result of Tesfahun and the author.

Traveling waves for a microscopic model of traffic flow
Wen Shen and Karim Shikh-Khalil
2018, 38(5): 2571-2589 doi: 10.3934/dcds.2018108 +[Abstract](29) +[HTML](14) +[PDF](496.15KB)

We consider the follow-the-leader model for traffic flow. The position of each car $z_i(t)$ satisfies an ordinary differential equation, whose speed depends only on the relative position $z_{i+1}(t)$ of the car ahead. Each car perceives a local density $ρ_i(t)$. We study a discrete traveling wave profile $W(x)$ along which the trajectory $(ρ_i(t), z_i(t))$ traces such that $W(z_i(t)) = ρ_i(t)$ for all $i$ and $t>0$; see definition 2.2. We derive a delay differential equation satisfied by such profiles. Existence and uniqueness of solutions are proved, for the two-point boundary value problem where the car densities at $x\to±∞$ are given. Furthermore, we show that such profiles are locally stable, attracting nearby monotone solutions of the follow-the-leader model.

Dynamics for the diffusive Leslie-Gower model with double free boundaries
Mingxin Wang and Qianying Zhang
2018, 38(5): 2591-2607 doi: 10.3934/dcds.2018109 +[Abstract](21) +[HTML](14) +[PDF](433.93KB)

In this paper we investigate a free boundary problem for the diffusive Leslie-Gower prey-predator model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species. We first prove the existence, uniqueness and regularity of global solution. Then provide a spreading-vanishing dichotomy, namely the predator species either successfully spreads to infinity as $t\to∞$ at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run. The long time behavior of $(u, v)$ and criteria for spreading and vanishing are also obtained. Because the term $v/u$ (which appears in the second equation) may be unbounded when $u$ nears zero, it will bring some difficulties for our study.

On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components
Wendong Wang, Liqun Zhang and Zhifei Zhang
2018, 38(5): 2609-2627 doi: 10.3934/dcds.2018110 +[Abstract](29) +[HTML](18) +[PDF](418.18KB)

We present some interior regularity criteria of the 3-D Navier-Stokes equations involving two components of the velocity. These results in particular imply that if the solution is singular at one point, then at least two components of the velocity have to blow up at the same point.

Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models
Zhijian Yang and Yanan Li
2018, 38(5): 2629-2653 doi: 10.3934/dcds.2018111 +[Abstract](35) +[HTML](26) +[PDF](496.62KB)

In this paper, we are concerned with the existence and stability of pullback exponential attractors for a non-autonomous dynamical system. (ⅰ) We propose two new criteria for the discrete dynamical system and continuous one, respectively. (ⅱ) By applying the criteria to the non-autonomous Kirchhoff wave models with structural damping and supercritical nonlinearity we construct a family of pullback exponential attractors which are stable with respect to perturbations.

Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system
Lei Zhang and Bin Liu
2018, 38(5): 2655-2685 doi: 10.3934/dcds.2018112 +[Abstract](37) +[HTML](21) +[PDF](581.01KB)

In this paper, we are concerned with the Cauchy problem for a new two-component Camassa-Holm system with the effect of the Coriolis force in the rotating fluid, which is a model in the equatorial water waves. We first investigate the local well-posedness of the system in \begin{document}$ B_{p,r}^s× B_{p,r}^{s-1}$\end{document} with \begin{document}$s>\max\{1+\frac{1}{p},\frac{3}{2},2-\frac{1}{p}\}$\end{document}, \begin{document}$p,r∈ [1,∞]$\end{document} by using the transport theory in Besov space. Then by means of the logarithmic interpolation inequality and the Osgood's lemma, we establish the local well-posedness in the critical Besov space \begin{document}$ B_{2,1}^{3/2}× B_{2,1}^{1/2}$\end{document}, and we present a blow-up result with the initial data in critical Besov space by virtue of the conservation law. Finally, we study the Gevrey regularity and analyticity of solutions to the system in a range of Gevrey-Sobolev spaces in the sense of Hardamard. Moreover, a precise lower bound of the lifespan is obtained.

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