# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

March 2020 , Volume 40 , Issue 3

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2020, 40(3): 1257-1281 doi: 10.3934/dcds.2020077 +[Abstract](1647) +[HTML](133) +[PDF](433.9KB)
Abstract:

In [5] Bourgain proves that Sarnak's disjointness conjecture holds for a certain class of three-interval exchange maps. In the present paper we slightly improve the Diophantine condition of Bourgain and estimate the constants in the proof. We further show that the new parameter set has positive, but not full Hausdorff dimension. This, in particular, implies that the Lebesgue measure of this set is zero.

2020, 40(3): 1283-1307 doi: 10.3934/dcds.2020078 +[Abstract](1786) +[HTML](146) +[PDF](429.84KB)
Abstract:

The global well-posedness for the KP-Ⅱ equation is established in the anisotropic Sobolev space \begin{document}$H^{s, 0}$\end{document} for \begin{document}$s>-\frac{3}{8}$\end{document}. Even though conservation laws are invalid in the Sobolev space with negative index, we explore the asymptotic behavior of the solution by the aid of the \begin{document}$I$\end{document}-method in which Colliander, Keel, Staffilani, Takaoka, and Tao introduced a series of modified energy terms. Moreover, a-priori estimate of the solution leads to the existence of global attractor for the weakly damped, forced KP-Ⅱ equation in the weak topology of the Sobolev space when \begin{document}$s>-\frac{1}{8}$\end{document}.

2020, 40(3): 1309-1360 doi: 10.3934/dcds.2020079 +[Abstract](1389) +[HTML](128) +[PDF](662.03KB)
Abstract:

We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation).

As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size \begin{document}$\delta$\end{document}, the physical measure varies continuously, with a modulus of continuity \begin{document}$O(\delta \log \delta )$\end{document}, which is asymptotically optimal for this kind of piecewise smooth maps.

2020, 40(3): 1361-1387 doi: 10.3934/dcds.2020080 +[Abstract](1613) +[HTML](125) +[PDF](471.56KB)
Abstract:

Young [17] proved the positivity of Lyapunov exponent in a large set of the energies for some quasi-periodic cocycles. Her result is also proved to be applicable for some quasi-periodic Schrödinger cocycles by Zhang [18]. However, her result cannot be applied to the Schrödinger cocycles with the potential \begin{document}$v = \cos(4\pi x)+w( x)$\end{document}, where \begin{document}$w\in C^2(\mathbb R/\mathbb Z,\mathbb R)$\end{document} is a small perturbation. In this paper, we will improve her result such that it can be applied to more cocycles.

2020, 40(3): 1389-1409 doi: 10.3934/dcds.2020081 +[Abstract](1414) +[HTML](95) +[PDF](399.14KB)
Abstract:

In this paper we will prove the existence of periodically invariant tori of twist mappings on resonant surfaces under the low dimensional intersection property.

2020, 40(3): 1411-1433 doi: 10.3934/dcds.2020082 +[Abstract](1748) +[HTML](108) +[PDF](391.25KB)
Abstract:

In this paper, we study the long-time existence and uniqueness (solvability) for the initial value problem of the 2D inviscid dispersive SQG equation. First we obtain the local solvability with existence-time independent of the amplitude parameter \begin{document}$A$\end{document}. Then, assuming more regularity and using a blow-up criterion of BKM type and a space-time estimate of Strichartz type, we prove long-time solvability of solutions in Besov spaces for large \begin{document}$A$\end{document} and arbitrary initial data. In comparison with previous results, we are able to consider improved cases of the regularity and larger initial data classes.

2020, 40(3): 1435-1492 doi: 10.3934/dcds.2020083 +[Abstract](1360) +[HTML](115) +[PDF](578.78KB)
Abstract:

For 2D compressible full Euler equations of Chaplygin gases, when the initial axisymmetric perturbation of a rest state is small, we prove that the smooth solution exists globally. Compared with the previous references, there are two different key points in this paper: both the vorticity and the variable entropy are simultaneously considered, moreover, the usual assumption on the compact support of initial perturbation is removed. Due to the appearances of the variable entropy and vorticity, the related perturbation of solution will have no decay in time, which leads to an essential difficulty in establishing the global energy estimate. Thanks to introducing a nonlinear ODE which arises from the vorticity and entropy, and considering the difference between the solutions of the resulting ODE and the full Euler equations, we can distinguish the fast decay part and non-decay part of solution to Euler equations. Based on this, by introducing some suitable weighted energies together with a class of weighted \begin{document}$L^\infty$\end{document}-\begin{document}$L^\infty$\end{document} estimates for the solutions of 2D wave equations, we can eventually obtain the global energy estimates and further complete the proof on the global existence of smooth solution to 2D full Euler equations.

2020, 40(3): 1493-1515 doi: 10.3934/dcds.2020084 +[Abstract](1560) +[HTML](146) +[PDF](379.52KB)
Abstract:

This paper is concerned with the stability and dynamics of a weak viscoelastic system with nonlinear time-varying delay. By imposing appropriate assumptions on the memory and sub-linear delay operator, we prove the global well-posedness and stability which generates a gradient system. The gradient system possesses finite fractal dimensional global and exponential attractors with unstable manifold structure. Moreover, the effect and balance between damping and time-varying delay are also presented.

2020, 40(3): 1517-1554 doi: 10.3934/dcds.2020085 +[Abstract](1187) +[HTML](90) +[PDF](517.88KB)
Abstract:

In this article, we consider the geodesic flow on a compact rank \begin{document}$1$\end{document} Riemannian manifold \begin{document}$M$\end{document} without focal points, whose universal cover is denoted by \begin{document}$X$\end{document}. On the ideal boundary \begin{document}$X(\infty)$\end{document} of \begin{document}$X$\end{document}, we show the existence and uniqueness of the Busemann density, which is realized via the Patterson-Sullivan measure. Based on the the Patterson-Sullivan measure, we show that the geodesic flow on \begin{document}$M$\end{document} has a unique invariant measure of maximal entropy. We also obtain the asymptotic growth rate of the volume of geodesic spheres in \begin{document}$X$\end{document} and the growth rate of the number of closed geodesics on \begin{document}$M$\end{document}. These results generalize the work of Margulis and Knieper in the case of negative and nonpositive curvature respectively.

2020, 40(3): 1555-1593 doi: 10.3934/dcds.2020086 +[Abstract](1100) +[HTML](88) +[PDF](487.61KB)
Abstract:

We consider a nonlinear, free boundary fluid-structure interaction model in a bounded domain. The viscous incompressible fluid interacts with a nonlinear elastic body on the common boundary via the velocity and stress matching conditions. The motion of the fluid is governed by incompressible Navier-Stokes equations while the displacement of elastic structure is determined by a nonlinear elastodynamic system with boundary dissipation. The boundary dissipation is inserted in the velocity matching condition. We prove the global existence of the smooth solutions for small initial data and obtain the exponential decay of the energy of this system as well.

2020, 40(3): 1595-1620 doi: 10.3934/dcds.2020087 +[Abstract](1102) +[HTML](131) +[PDF](624.8KB)
Abstract:

We consider a system of coupled cubic Schrödinger equations in one space dimension

in the non-integrable case \begin{document}$0 < \omega < 1$\end{document}.

First, we justify the existence of a symmetric 2-solitary wave with logarithmic distance, i.e. a solution of the system satisfying

where \begin{document}$Q = \sqrt{2}$\end{document}sech is the explicit solution of \begin{document}$Q'' - Q + Q^3 = 0$\end{document} and \begin{document}$\Omega>0$\end{document} is a constant. This result extends to the non-integrable case the existence of symmetric 2-solitons with logarithmic distance known in the integrable case \begin{document}$\omega = 0$\end{document} and \begin{document}$\omega = 1$\end{document} ([15,33]). Such strongly interacting symmetric \begin{document}$2$\end{document}-solitary waves were also previously constructed for the non-integrable scalar nonlinear Schrödinger equation in any space dimension and for any energy-subcritical power nonlinearity ([20,22]).

Second, under the conditions \begin{document}$0<c<1$\end{document} and \begin{document}$0<\omega < \frac 12 c(c+1)$\end{document}, we construct solutions of the system satisfying

where \begin{document}$Q_c(x) = cQ(cx)$\end{document} and \begin{document}$\Omega_c>0$\end{document} is a constant. Such logarithmic regime with non-symmetric solitons does not exist in the integrable cases \begin{document}$\omega = 0$\end{document} and \begin{document}$\omega = 1$\end{document} and is still unknown in the non-integrable scalar case.

2020, 40(3): 1621-1663 doi: 10.3934/dcds.2020088 +[Abstract](1146) +[HTML](113) +[PDF](5347.37KB)
Abstract:

We study the permanence and impermanence for discrete-time Kolmogorov systems admitting a carrying simplex. Sufficient conditions to guarantee permanence and impermanence are provided based on the existence of a carrying simplex. Particularly, for low-dimensional systems, permanence and impermanence can be determined by boundary fixed points. For a class of competitive systems whose fixed points are determined by linear equations, there always exists a carrying simplex. We provide a universal classification via the equivalence relation relative to local dynamics of boundary fixed points for the three-dimensional systems by the index formula on the carrying simplex. There are a total of \begin{document}$33$\end{document} stable equivalence classes which are described in terms of inequalities on parameters, and we present the phase portraits on their carrying simplices. Moreover, every orbit converges to some fixed point in classes \begin{document}$1-25$\end{document} and \begin{document}$33$\end{document}; there is always a heteroclinic cycle in class \begin{document}$27$\end{document}; Neimark-Sacker bifurcations may occur in classes \begin{document}$26-31$\end{document} but cannot occur in class \begin{document}$32$\end{document}. Based on our permanence criteria and the equivalence classification, we obtain the specific conditions on parameters for permanence and impermanence. Only systems in classes \begin{document}$29, 31, 33$\end{document} and those in class \begin{document}$27$\end{document} with a repelling heteroclinic cycle are permanent. Applications to discrete population models including the Leslie-Gower models, Atkinson-Allen models and Ricker models are given.

2020, 40(3): 1665-1702 doi: 10.3934/dcds.2020089 +[Abstract](1263) +[HTML](147) +[PDF](406.9KB)
Abstract:

The variation-of-constants formula is one of the principal tools of the theory of differential equations. There are many papers dealing with different versions of the variation-of-constants formula and its applications. Our main purpose in this paper is to give a variation-of-constants formula for inhomogeneous linear functional differential systems determined by general Volterra type operators with delay. Our treatment of the delay in the considered systems is completely different from the usual methods. We deal with the representation of the studied Volterra type operators. Some existence and uniqueness theorems are obtained for the studied linear functional differential and integral systems. Finally, some applications are given.

2020, 40(3): 1703-1735 doi: 10.3934/dcds.2020090 +[Abstract](1483) +[HTML](101) +[PDF](543.08KB)
Abstract:

The orbital stability of peakons and hyperbolic periodic peakons for the Camassa-Holm equation has been established by Constantin and Strauss in [A. Constantin, W. Strauss, Comm. Pure. Appl. Math. 53 (2000) 603-610] and Lenells in [J. Lenells, Int. Math. Res. Not. 10 (2004) 485-499], respectively. In this paper, we prove the orbital stability of the elliptic periodic peakons for the modified Camassa-Holm equation. By using the invariants of the equation and controlling the extrema of the solution, it is demonstrated that the shapes of these elliptic periodic peakons are stable under small perturbations in the energy space. Throughout the paper, the theory of elliptic functions and elliptic integrals is used in the calculation.

2020, 40(3): 1737-1755 doi: 10.3934/dcds.2020091 +[Abstract](1255) +[HTML](152) +[PDF](403.52KB)
Abstract:

This paper deals with the quasilinear parabolic-elliptic chemotaxis system

under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$\Omega\subset\mathbb{R}^{n}$\end{document} with smooth boundary, where \begin{document}$\tau\in\{0, 1\}$\end{document}, \begin{document}$\chi>0$\end{document}, \begin{document}$\mu>0$\end{document} and \begin{document}$r\geq2$\end{document}. \begin{document}$D(u)$\end{document} is supposed to satisfy

It is shown that when \begin{document}$\mu>\frac{\chi^{2}}{16}$\end{document} and \begin{document}$r\geq2$\end{document}, then the solution to the system exponentially converges to the constant stationary solution \begin{document}$(1, 1)$\end{document}.

2020, 40(3): 1757-1774 doi: 10.3934/dcds.2020092 +[Abstract](1285) +[HTML](137) +[PDF](408.58KB)
Abstract:

In this paper we consider a von Karman plate equation with memory-type boundary conditions. By assuming the relaxation function \begin{document}$k_i$\end{document} \begin{document}$(i = 1, 2)$\end{document} with minimal conditions on the \begin{document}$L^1(0, \infty)$\end{document}, we establish an optimal explicit and general energy decay result. In particular, the energy result holds for \begin{document}$H(s) = s^p$\end{document} with the full admissible range \begin{document}$[1, 2)$\end{document} instead of \begin{document}$[1, 3/2)$\end{document}. This result is new and substantially improves earlier results in the literature.

2020, 40(3): 1775-1798 doi: 10.3934/dcds.2020093 +[Abstract](1842) +[HTML](95) +[PDF](387.51KB)
Abstract:

We study the low-energy solutions to the 3D compressible Navier-Stokes-Poisson equations. We first obtain the existence of smooth solutions with small \begin{document}$L^2$\end{document}-norm and essentially bounded densities. No smallness assumption is imposed on the \begin{document}$H^4$\end{document}-norm of the initial data. Using a compactness argument, we further obtain the existence of weak solutions which may have discontinuities across some hypersurfaces in \begin{document}$\mathbb R^3$\end{document}. We also provide a blow-up criterion of solutions in terms of the \begin{document}$L^\infty$\end{document}-norm of density.

2020, 40(3): 1799-1811 doi: 10.3934/dcds.2020094 +[Abstract](1339) +[HTML](98) +[PDF](387.88KB)
Abstract:

For \begin{document}$n \ge 1$\end{document}, consider the space of affine conjugacy classes of topological cubic polynomials \begin{document}$f: \mathbb{C} \to \mathbb{C}$\end{document} with a period \begin{document}$n$\end{document} ramification point. It is shown that this space is a connected topological space.

2020, 40(3): 1813-1846 doi: 10.3934/dcds.2020095 +[Abstract](1412) +[HTML](96) +[PDF](502.87KB)
Abstract:

In this note, we use an epiperimetric inequality approach to study the regularity of the free boundary for the parabolic Signorini problem. We show that if the "vanishing order" of a solution at a free boundary point is close to \begin{document}$3/2$\end{document} or an even integer, then the solution is asymptotically homogeneous. Furthermore, one can derive a convergence rate estimate towards the asymptotic homogeneous solution. As a consequence, we obtain the regularity of the regular free boundary as well as the frequency gap.

2020, 40(3): 1847-1878 doi: 10.3934/dcds.2020096 +[Abstract](1509) +[HTML](100) +[PDF](445.15KB)
Abstract:

We consider a coupled bulk–surface Allen–Cahn system affixed with a Robin-type boundary condition between the bulk and surface variables. This system can also be viewed as a relaxation to a bulk–surface Allen–Cahn system with non-trivial transmission conditions. Assuming that the nonlinearities are real analytic, we prove the convergence of every global strong solution to a single equilibrium as time tends to infinity. Furthermore, we obtain an estimate on the rate of convergence. The proof relies on an extended Łojasiewicz–Simon type inequality for the bulk–surface coupled system. Compared with previous works, new difficulties arise as in our system the surface variable is no longer the trace of the bulk variable, but now they are coupled through a nonlinear Robin boundary condition.

2020, 40(3): 1879-1887 doi: 10.3934/dcds.2020097 +[Abstract](1491) +[HTML](100) +[PDF](311.3KB)
Abstract:

In 3-dimensional manifolds, we prove that generically in \begin{document}$\operatorname{Diff}^{1}_{m}(M^{3})$\end{document}, the existence of a minimal expanding invariant foliation implies stable Bernoulliness.

2020, 40(3): 1889-1902 doi: 10.3934/dcds.2020098 +[Abstract](1394) +[HTML](118) +[PDF](367.04KB)
Abstract:

In this paper we study the existence and some properties of the global attractors for a class of weighted equations when the weighted Sobolev space \begin{document}$H_0^{1,a}(\Omega)$\end{document} (see Definition 1.1) cannot be bounded embedded into \begin{document}$L^2(\Omega)$\end{document}. We claim that the dimension of the global attractor is infinite by estimating its lower bound of \begin{document}$Z_2$\end{document}-index. Moreover, we prove that there is an infinite sequence of stationary points in the global attractor which goes to 0 and the corresponding critical value sequence of the Lyapunov functional also goes to 0.

2020, 40(3): 1903-1935 doi: 10.3934/dcds.2020099 +[Abstract](1373) +[HTML](101) +[PDF](413.2KB)
Abstract:

In this paper, we consider the Landau-Lifshitz-Gilbert systems with spin-polarized transport from a bounded domain in \begin{document}$\mathbb{R}^3$\end{document} into \begin{document}$S^2$\end{document} and show the existence of global weak solutions to the Cauchy problems of such Landau-Lifshtiz systems. In particular, we show that the Cauchy problem to Landau-Lifshitz equation without damping but with diffusion process of the spin accumulation admits a global weak solution. The Landau-Lifshtiz system with spin-polarized transport into a compact Lie algebra is also discussed and some similar results are proved. The key ingredients of this article consist of the choices of test functions and approximate equations.

2020, 40(3): 1937-1961 doi: 10.3934/dcds.2020100 +[Abstract](1233) +[HTML](100) +[PDF](418.96KB)
Abstract:

This paper is concerned with the long-time behavior for a class of non-autonomous plate equations with perturbation and strong damping of \begin{document}$p$\end{document}-Laplacian type

in bounded domain \begin{document}$\Omega\subset \mathbb{R}^N$\end{document} with smooth boundary and critical nonlinear terms. The global existence of weak solution which generates a continuous process has been presented firstly, then the existence of strong and weak uniform attractors with non-compact external forces also derived. Moreover, the upper-semicontinuity of uniform attractors under small perturbations has also obtained by delicate estimate and contradiction argument.

2020, 40(3): 1963-1987 doi: 10.3934/dcds.2020101 +[Abstract](1233) +[HTML](145) +[PDF](408.62KB)
Abstract:

A vanishing viscosity problem for the Patlak-Keller-Segel model is studied in this paper. This is a parabolic-parabolic system in a bounded domain \begin{document}$\Omega\subset \mathbb{R}^n$\end{document}, with a vanishing viscosity \begin{document}$\varepsilon\to 0$\end{document}. We show that if the initial value lies in \begin{document}$W^{1, p}$\end{document} with \begin{document}$p>\max\{2, n\}$\end{document}, then there exists a unique solution \begin{document}$(u_\varepsilon, v_\varepsilon)$\end{document} with its lifespan independent of \begin{document}$\varepsilon$\end{document}. Furthermore, as \begin{document}$\varepsilon\rightarrow 0$\end{document}, \begin{document}$(u_\varepsilon, v_\varepsilon)$\end{document} converges to the solution \begin{document}$(u, v)$\end{document} of the limiting system in a suitable sense.

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