# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

September 2019 , Volume 39 , Issue 9

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2019, 39(9): 4929-4943 doi: 10.3934/dcds.2019201 +[Abstract](1650) +[HTML](229) +[PDF](310.27KB)
Abstract:

In this paper we study the asymptotic behavior of a very fast diffusion PDE in 1D with periodic boundary conditions. This equation is motivated by the gradient flow approach to the problem of quantization of measures introduced in [3]. We prove exponential convergence to equilibrium under minimal assumptions on the data, and we also provide sufficient conditions for \begin{document}$W_2$\end{document}-stability of solutions.

2019, 39(9): 4945-4953 doi: 10.3934/dcds.2019202 +[Abstract](1655) +[HTML](205) +[PDF](298.14KB)
Abstract:

After investigating existence and uniqueness of the global strong solutions for Euler-Voigt equations under Dirichlet conditions, we obtain the Kato's type theorems for the convergence of the Euler-Voigt equations to Euler equations. More precisely, the necessary and sufficient conditions that the solution of Euler-Voigt equation converges to the one of Euler equations, as \begin{document}$\alpha\to 0$\end{document}, can be obtained.

2019, 39(9): 4955-4977 doi: 10.3934/dcds.2019203 +[Abstract](1747) +[HTML](224) +[PDF](538.6KB)
Abstract:

We investigate the statistical properties of piecewise expanding maps on the unit interval, whose inverse Jacobian may have low regularity near singularities. The method is new yet simple: instead of directly working with the 1-d map, we first lift the 1-d expanding map to a hyperbolic map on the unit square, and then take advantage of the functional analytic method developed by Demers and Zhang in [21,22,23] for hyperbolic systems with singularities. By projecting back to the 1-d map, we are able to prove that it inherits nice statistical properties, including the large deviation principle, the exponential decay of correlations, as well as the almost sure invariance principle for the expanding map on a large class of observables. Moreover, we are able to prove that the projected SRB measure has a piecewise continuous density function. Our results apply to rather general 1-d expanding maps, including some \begin{document}$C^1$\end{document} perturbations of the Lorenz-like map and the Gauss map whose statistical properties are still unknown as they fail all other available methods.

2019, 39(9): 4979-5015 doi: 10.3934/dcds.2019204 +[Abstract](32971) +[HTML](192) +[PDF](498.73KB)
Abstract:

In this paper, we continue our study of the Boltzmann equation by use of tools originating from the analysis of dispersive equations in quantum dynamics. Specifically, we focus on properties of solutions to the Boltzmann equation with collision kernel equal to a constant in the spatial domain \begin{document}$\mathbb{R}^d$\end{document}, \begin{document}$d\geq 2$\end{document}, which we use as a model in this paper. Local well-posedness for this equation has been proven using the Wigner transform when \begin{document}$\left< v \right>^\beta f_0 \in L^2_v H^\alpha_x$\end{document} for \begin{document}$\min (\alpha,\beta) > \frac{d-1}{2}$\end{document}. We prove that if \begin{document}$\alpha,\beta$\end{document} are large enough, then it is possible to propagate moments in \begin{document}$x$\end{document} and derivatives in \begin{document}$v$\end{document} (for instance, \begin{document}$\left< x \right>^k \left< \nabla_v \right>^\ell f \in L^\infty_T L^2_{x,v}$\end{document} if \begin{document}$f_0$\end{document} is nice enough). The mechanism is an exchange of regularity in return for moments of the (inverse) Wigner transform of \begin{document}$f$\end{document}. We also prove a persistence of regularity result for the scale of Sobolev spaces \begin{document}$H^{\alpha,\beta}$\end{document}; and, continuity of the solution map in \begin{document}$H^{\alpha,\beta}$\end{document}. Altogether, these results allow us to conclude non-negativity of solutions, conservation of energy, and the \begin{document}$H$\end{document}-theorem for sufficiently regular solutions constructed via the Wigner transform. Non-negativity in particular is proven to hold in \begin{document}$H^{\alpha,\beta}$\end{document} for any \begin{document}$\alpha,\beta > \frac{d-1}{2}$\end{document}, without any additional regularity or decay assumptions.

2019, 39(9): 5017-5083 doi: 10.3934/dcds.2019205 +[Abstract](1837) +[HTML](272) +[PDF](789.61KB)
Abstract:

We establish the existence and nonlinear stability of travelling pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions close to the continuum limit. For the verification of the spectral properties, we need to study a functional differential equation of mixed type (MFDE) with unbounded shifts. We avoid the use of exponential dichotomies and phase spaces, by building on a technique developed by Bates, Chen and Chmaj for the discrete Nagumo equation. This allows us to transfer several crucial Fredholm properties from the PDE setting to our discrete setting.

2019, 39(9): 5085-5103 doi: 10.3934/dcds.2019206 +[Abstract](1682) +[HTML](188) +[PDF](377.43KB)
Abstract:

We prove that every topologically transitive shift of finite type in one dimension is topologically conjugate to a subshift arising from a primitive random substitution on a finite alphabet. As a result, we show that the set of values of topological entropy which can be attained by random substitution subshifts contains the logarithm of all Perron numbers and so is dense in the positive real numbers. We also provide an independent proof of this density statement using elementary methods.

2019, 39(9): 5105-5124 doi: 10.3934/dcds.2019207 +[Abstract](1502) +[HTML](192) +[PDF](365.31KB)
Abstract:

In recent collaborative work we studied existence and uniqueness of a Lagrangian description for absolutely continuous curves in spaces of Borel probabilities on the real line with finite moments of given order. Of course, a measurable velocity driving the evolution in Eulerian coordinates is necessary to define the Eulerian and Lagrangian descriptions of fluid flow; here we prove that in this case it is also sufficient for a Lagrangian description. More precisely, we argue that the existence of the integrable velocity along an absolutely continuous curve in the set of Borel probabilities on the line is enough to produce a canonical Lagrangian description for the curve; this is given by the family of optimal maps between the uniform distribution on the unit interval and the measures on the curve. Moreover, we identify a necessary and sufficient condition on said family of optimal maps which ensures that the measurable velocity along the curve exists.

2019, 39(9): 5125-5147 doi: 10.3934/dcds.2019208 +[Abstract](1442) +[HTML](206) +[PDF](502.76KB)
Abstract:

In this paper we show how solutions of a wave equation with distributed damping near the boundary converge to solutions of a wave equation with boundary feedback damping. Sufficient conditions are given for the convergence of solutions to occur in the natural energy space.

2019, 39(9): 5149-5169 doi: 10.3934/dcds.2019209 +[Abstract](1599) +[HTML](209) +[PDF](446.01KB)
Abstract:

The aim of this paper is to obtain the dispersion relation for small-amplitude periodic travelling water waves propagating over a flat bed with a specified mean depth under the presence of a discontinuous piecewise constant vorticity. An analysis of the dispersion relation for a model with two rotational layers each having a non-zero constant vorticity is presented. Moreover, we present a stability result for the bifurcation inducing laminar flow solutions.

2019, 39(9): 5171-5183 doi: 10.3934/dcds.2019210 +[Abstract](1399) +[HTML](185) +[PDF](357.19KB)
Abstract:

We present a new exact and explicit Pollard-like solution describing internal water waves representing the oscillation of the thermocline in a nonhydrostatic model. The derived solution is a modification of Pollard's surface wave solution in order to describe internal water waves at general latitudes. The novelty of this model consists in the embodiment of transitional layers beneath the thermocline. We present a Lagrangian analysis of the nonlinear internal water waves and we show the existence of two modes of the wave motion.

2019, 39(9): 5185-5206 doi: 10.3934/dcds.2019211 +[Abstract](1645) +[HTML](244) +[PDF](817.06KB)
Abstract:

In this paper, we investigate the dynamics of the following family of rational maps

with one parameter \begin{document}$\lambda \in \mathbb{C}^* - \{\lambda: \lambda^{2n + 2} = 1\}$\end{document}, where \begin{document}$n\geq 2$\end{document}. This family of rational maps is viewed as a singular perturbation of the bi-critical map \begin{document}$P_{-n}(z) = z^{-n}$\end{document} if \begin{document}$\lambda \neq 0$\end{document} is small. It is proved that the Julia set \begin{document}$J(f_\lambda)$\end{document} is either a quasicircle, a Cantor set of circles, a Sierpiński carpet or a degenerate Sierpiński carpet provided the free critical orbits of \begin{document}$f_\lambda$\end{document} are attracted by the super-attracting cycle \begin{document}$0\leftrightarrow\infty$\end{document}. Furthermore, we prove that there exists suitable \begin{document}$\lambda$\end{document} such that \begin{document}$J(f_\lambda)$\end{document} is a Cantor set of circles but the dynamics of \begin{document}$f_{\lambda}$\end{document} on \begin{document}$J(f_{\lambda})$\end{document} is not topologically conjugate to that of any known rational maps with only one or two free critical orbits (including McMullen maps and the generalized McMullen maps). The connectivity of \begin{document}$J(f_{\lambda})$\end{document} is also proved if the free critical orbits are not attracted by the cycle \begin{document}$0\leftrightarrow\infty$\end{document}. Finally we give an estimate of the Hausdorff dimension of the Julia set of \begin{document}$f_\lambda$\end{document} in some special cases.

2019, 39(9): 5207-5222 doi: 10.3934/dcds.2019212 +[Abstract](1692) +[HTML](189) +[PDF](383.8KB)
Abstract:

Moser-Trudinger inequality was generalised by Calanchi-Ruf to the following version: If \begin{document}$\beta \in [0,1)$\end{document} and \begin{document}$w_0(x) = |\log |x||^{\beta(n-1)}$\end{document} or \begin{document}$\left( \log \frac{e}{|x|}\right)^{\beta(n-1)}$\end{document} then

if and only if \begin{document}$\alpha \leq \alpha_\beta = n\left[\omega_{n-1}^{\frac{1}{n-1}}(1-\beta) \right]^{\frac{1}{1-\beta}}$\end{document} where \begin{document}$\omega_{n-1}$\end{document} denotes the surface measure of the unit sphere in \begin{document}$\mathbb {R}^n$\end{document}. The primary goal of this work is to address the issue of existence of extremal function for the above inequality. A non-existence (of extremal function) type result is also discussed, for the usual Moser-Trudinger functional.

2019, 39(9): 5223-5262 doi: 10.3934/dcds.2019213 +[Abstract](1760) +[HTML](195) +[PDF](522.24KB)
Abstract:

This paper is concerned with a diffusive Lotka-Volterra type competition system with a free boundary in one space dimension. Such a system may be used to describe the invasion of a new species into the habitat of a native competitor, and its long-time dynamical behavior can be described by a spreading-vanishing dichotomy. The main purpose of this paper is to determine the asymptotic spreading speed of the invading species when its spreading is successful, which involves two systems of traveling wave type equations.

2019, 39(9): 5263-5273 doi: 10.3934/dcds.2019214 +[Abstract](1616) +[HTML](193) +[PDF](365.85KB)
Abstract:

In this paper, we consider \begin{document}$L^2$\end{document} constrained minimization problem for a modified Gross-Pitaevskii equation with higher order interactions in \begin{document}$\mathbb{R}^2$\end{document}. By using an auxiliary functional and some detailed energy estimates, the blow-up behavior of ground state for the modified Gross-Pitaevskii equation was obtained under different parameter regimes. Our conclusion extends some results of [3,Theorem 3.4].

2019, 39(9): 5275-5299 doi: 10.3934/dcds.2019215 +[Abstract](2280) +[HTML](269) +[PDF](1115.98KB)
Abstract:

In this article, we prove the existence of a saddle-node bifurcation of limit cycles in continuous piecewise linear systems with three zones. The bifurcation arises from the perturbation of a non-generic situation, where there exists a linear center in the middle zone. We obtain an approximation of the relation between the parameters of the system, such that the saddle-node bifurcation takes place, as well as of the period and amplitude of the non-hyperbolic limit cycle that bifurcates. We consider two applications, first a piecewise linear version of the FitzHugh-Nagumo neuron model of spike generation and second an electronic circuit, the memristor oscillator.

2019, 39(9): 5301-5317 doi: 10.3934/dcds.2019216 +[Abstract](1440) +[HTML](186) +[PDF](378.24KB)
Abstract:

In this paper we study the topological characteristic factors along cubes of minimal systems. It is shown that up to proximal extensions the pro-nilfactors are the topological characteristic factors along cubes of minimal systems. In particular, for a distal minimal system, the maximal \begin{document}$(d-1)$\end{document}-step pro-nilfactor is the topological cubic characteristic factor of order \begin{document}$d$\end{document}.

2019, 39(9): 5319-5337 doi: 10.3934/dcds.2019217 +[Abstract](1369) +[HTML](193) +[PDF](410.68KB)
Abstract:

We prove that for any infinite collection of quadratic Julia sets, there exists a transcendental entire function whose Julia set contains quasiconformal copies of the given quadratic Julia sets. In order to prove the result, we construct a quasiregular map with required dynamics and employ the quasiconformal surgery to obtain the desired transcendental entire function. In addition, the transcendental entire function has order zero.

2019, 39(9): 5339-5363 doi: 10.3934/dcds.2019218 +[Abstract](1588) +[HTML](182) +[PDF](443.84KB)
Abstract:

Contraction analysis uses a local criterion to prove the long-term behaviour of a dynamical system. A contraction metric is a Riemannian metric with respect to which the distance between adjacent solutions contracts. If adjacent solutions in all directions perpendicular to the flow are contracted, then there exists a unique periodic orbit, which is exponentially stable and we obtain an upper bound on the rate of exponential attraction.

In this paper we study the converse question and show that, given an exponentially stable periodic orbit, a contraction metric exists on its basin of attraction and we can recover the upper bound on the rate of exponential attraction.

2019, 39(9): 5365-5402 doi: 10.3934/dcds.2019220 +[Abstract](1914) +[HTML](226) +[PDF](479.97KB)
Abstract:

We consider the Cauchy problem of a Keller-Segel type chemotaxis model with logarithmic sensitivity and logistic growth. We study the global well-posedness, long-time behavior, vanishing coefficient limit and decay rate of solutions in \begin{document}$\mathbb{R}$\end{document}. By utilizing energy methods, we show that for any given classical initial datum which is a perturbation around a constant equilibrium state with finite energy (not small), there exists a unique global-in-time solution to the Cauchy problem, and the solution converges to the constant equilibrium state, as time goes to infinity. Under the same initial condition, it is shown that the solution with positive chemical diffusion coefficient converges to the solution with zero chemical diffusion coefficient, as the coefficient goes to zero. Furthermore, for a slightly smaller class of initial data, we identify the algebraic decay rates of the solution to the constant equilibrium state by employing time-weighted energy estimates.

2019, 39(9): 5403-5429 doi: 10.3934/dcds.2019221 +[Abstract](1618) +[HTML](175) +[PDF](456.59KB)
Abstract:

In this paper, we study stochastic homogenization of a coupled diffusion-reaction system. The diffusion-reaction system is coupled to stochastic differential equations, which govern the changes in the media properties. Though homogenization with changing media properties has been studied in previous findings, there is little research on homogenization when the media properties change due to stochastic differential equations. Such processes occur in many applications, where the changes in media properties are due to particle deposition. In the paper, we investigate the well-posedness of the nonlinear fine-grid (resolved) problem and derive limiting equations. We formulate the cell problems and derive the limiting equations, which are deterministic with nonlinear reaction terms. The limiting equations involve the invariant measures corresponding to stochastic differential equations. The obtained results can play an important role for modeling in porous media and allow the use of simplified and deterministic limiting equations.

2019, 39(9): 5431-5463 doi: 10.3934/dcds.2019222 +[Abstract](1582) +[HTML](178) +[PDF](524.85KB)
Abstract:

The present paper is a continuation of our recent paper [4]. We will consider the following Cauchy problem for semi-linear structurally damped \begin{document}$\sigma$\end{document}-evolution models:

with \begin{document}$\sigma \ge 1$\end{document}, \begin{document}$\mu>0$\end{document} and \begin{document}$\delta \in (\frac{\sigma}{2}, \sigma]$\end{document}. Our aim is to study two main models including \begin{document}$\sigma$\end{document}-evolution models with structural damping \begin{document}$\delta \in (\frac{\sigma}{2}, \sigma)$\end{document} and those with visco-elastic damping \begin{document}$\delta = \sigma$\end{document}. Here the function \begin{document}$f(u, u_t)$\end{document} stands for power nonlinearities \begin{document}$|u|^{p}$\end{document} and \begin{document}$|u_t|^{p}$\end{document} with a given number \begin{document}$p>1$\end{document}. We are interested in investigating the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on \begin{document}$L^q$\end{document} spaces by assuming additional \begin{document}$L^{m}$\end{document} regularity for the initial data, with \begin{document}$q\in (1, \infty)$\end{document} and \begin{document}$m\in [1, q)$\end{document}.

2019, 39(9): 5465-5489 doi: 10.3934/dcds.2019223 +[Abstract](2043) +[HTML](193) +[PDF](1975.2KB)
Abstract:

In this paper, we study the emergent behaviors of the Cucker-Smale (C-S) ensemble under the interplay of memory effect and flocking dynamics. As a mathematical model incorporating aforementioned interplay, we introduce the fractional C-S model which can be obtained by replacing the usual time derivative by the Caputo fractional time derivative. For the proposed fractional C-S model, we provide a sufficient framework which admits the emergence of anomalous flocking with the algebraic decay and an \begin{document}$\ell^2$\end{document}-stability estimate with respect to initial data. We also provide several numerical examples and compare them with our theoretical results.

2019, 39(9): 5491-5520 doi: 10.3934/dcds.2019224 +[Abstract](1468) +[HTML](179) +[PDF](456.12KB)
Abstract:

We study some already introduced and some new strong and weak topologies of integral type to provide continuous dependence on continuous initial data for the solutions of non-autonomous Carathéodory delay differential equations. As a consequence, we obtain new families of continuous skew-product semiflows generated by delay differential equations whose vector fields belong to such metric topological vector spaces of Lipschitz Carathéodory functions. Sufficient conditions for the equivalence of all or some of the considered strong or weak topologies are also given. Finally, we also provide results of continuous dependence of the solutions as well as of continuity of the skew-product semiflows generated by Carathéodory delay differential equations when the considered phase space is a Sobolev space.

2019, 39(9): 5521-5541 doi: 10.3934/dcds.2019225 +[Abstract](1505) +[HTML](187) +[PDF](887.65KB)
Abstract:

In the present study we consider three two-component (integrable and non-integrable) systems which describe the propagation of shallow water waves on a constant shear current. Namely, we consider the two-component Camassa–Holm equations, the Zakharov–Ito system and the Kaup–Boussinesq equations all including constant vorticity effects. We analyze both solitary and periodic-type travelling waves using the simple and geometrically intuitive phase space analysis. We get the pulse-type solitary wave solutions and the front solitary wave solutions. For the Zakharov–Ito system we underline the occurrence of the pulse and anti-pulse solutions. The front wave solutions decay algebraically in the far field. For the Kaup–Boussinesq system, interesting analytical multi-pulsed travelling wave solutions are found.

2019  Impact Factor: 1.338