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Discrete & Continuous Dynamical Systems - A

November 2018 , Volume 38 , Issue 11

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Liouville theorems and classification results for a nonlocal Schrödinger equation
Yutian Lei
2018, 38(11): 5351-5377 doi: 10.3934/dcds.2018236 +[Abstract](546) +[HTML](77) +[PDF](542.95KB)
Abstract:

In this paper, we study the existence and the nonexistence of positive classical solutions of the static Hartree-Poisson equation

where \begin{document}$n ≥ 3$\end{document} and \begin{document}$p≥ 1$\end{document}. The exponents of the Serrin type, the Sobolev type and the Joseph-Lundgren type play the critical roles as in the study of the Lane-Emden equation. First, we prove that the equation has no positive solution when \begin{document}$1 ≤ p <\frac{n+2}{n-2}$\end{document} by means of the method of moving planes to the following system

When \begin{document}$p = \frac{n+2}{n-2}$\end{document}, all the positive solutions can be classified as

with the help of an integral system involving the Newton potential, where \begin{document}$c, t$\end{document} are positive constants, and \begin{document}$x^* ∈ R^n$\end{document}. In addition, we also give other equivalent conditions to classify those positive solutions. When \begin{document}$p>\frac{n+2}{n-2}$\end{document}, by the shooting method and the Pohozaev identity, we find radial solutions for the system. In particular, the equation has a radial solution decaying with slow rate \begin{document}$\frac{2}{p-1}$\end{document}. Finally, we point out that the equation has positive stable solutions if and only if \begin{document}$p ≥ 1+\frac{4}{n-4-2\sqrt{n-1}}$\end{document}.

Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus
Pedro Duarte and Silvius Klein
2018, 38(11): 5379-5387 doi: 10.3934/dcds.2018237 +[Abstract](176) +[HTML](52) +[PDF](352.57KB)
Abstract:

Consider the space of analytic, quasi-periodic cocycles on the higher dimensional torus. We provide examples of cocycles with nontrivial Lyapunov spectrum, whose homotopy classes do not contain any cocycles satisfying the dominated splitting property. This shows that the main result in the recent work "Complex one-frequency cocycles" by A. Avila, S. Jitomirskaya and C. Sadel does not hold in the higher dimensional torus setting.

On the concentration of semiclassical states for nonlinear Dirac equations
Xu Zhang
2018, 38(11): 5389-5413 doi: 10.3934/dcds.2018238 +[Abstract](264) +[HTML](64) +[PDF](505.8KB)
Abstract:

In this paper, we study the following nonlinear Dirac equation

where \begin{document}$a > 0$\end{document} is a constant, \begin{document}$α = (α_1, α_2, α_3)$\end{document}, \begin{document}$α_1, α_2, α_3$\end{document} and \begin{document}$β$\end{document} are \begin{document}$4×4$\end{document} Pauli-Dirac matrices. Under the assumptions that \begin{document}$V$\end{document} and \begin{document}$g$\end{document} are continuous but are not necessarily of class \begin{document}$C^1$\end{document}, when \begin{document}$g$\end{document} is super-linear growth at infinity we obtain the existence of semiclassical solutions, which converge to the least energy solutions of its limit problem as \begin{document}$\varepsilon \to 0$\end{document}.

On a new two-component $b$-family peakon system with cubic nonlinearity
Kai Yan, Zhijun Qiao and Yufeng Zhang
2018, 38(11): 5415-5442 doi: 10.3934/dcds.2018239 +[Abstract](205) +[HTML](56) +[PDF](1748.79KB)
Abstract:

In this paper, we propose a two-component \begin{document}$b$\end{document}-family system with cubic nonlinearity and peaked solitons (peakons) solutions, which includes the celebrated Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation and its two-component extension as special cases. Firstly, we study single peakon and multi-peakon solutions to the system. Then the local well-posedness for the Cauchy problem of the system is discussed. Furthermore, we derive the precise blow-up scenario and global existence for strong solutions to the two-component \begin{document}$b$\end{document}-family system with cubic nonlinearity. Finally, we investigate the asymptotic behaviors of strong solutions at infinity within its lifespan provided the initial data decay exponentially and algebraically.

Periodic solutions for the N-vortex problem via a superposition principle
Björn Gebhard
2018, 38(11): 5443-5460 doi: 10.3934/dcds.2018240 +[Abstract](111) +[HTML](46) +[PDF](1068.5KB)
Abstract:

We examine the \begin{document} $N$ \end{document}-vortex problem on general domains \begin{document} $Ω\subset\mathbb{R}^2$ \end{document} concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system of the form

where \begin{document} $Γ_k∈\mathbb{R}\setminus\{0\}$ \end{document} is the strength of the \begin{document} $k$ \end{document}th vortex at position \begin{document} $z_k(t)∈Ω$ \end{document}, \begin{document} $J∈\mathbb{R}^{2× 2}$ \end{document} is the standard symplectic matrix and

with some regular and symmetric, but in general not explicitely known function \begin{document} $g:Ω×Ω \to \mathbb{R}$ \end{document}. The investigation relies on the idea to superpose a stationary solution of a system of less than \begin{document} $N$ \end{document} vortices and several clusters of vortices that are close to rigidly rotating configurations of the whole-plane system. We establish general conditions on both, the stationary solution and the configurations, under which multiple \begin{document} $T$ \end{document}-periodic solutions are shown to exist for every \begin{document} $T>0$ \end{document} small enough. The crucial condition holds in generic bounded domains and is explicitly verified for an example in the unit disc \begin{document} $Ω = B_1(0)$ \end{document}. In particular we therefore obtain various examples of periodic solutions in \begin{document} $B_1(0)$ \end{document} that are not rigidly rotating configurations.

Positive solutions for a nonlinear Schrödinger-Poisson system
Chunhua Wang and Jing Yang
2018, 38(11): 5461-5504 doi: 10.3934/dcds.2018241 +[Abstract](164) +[HTML](49) +[PDF](631.71KB)
Abstract:

In this paper, we study the following nonlinear Schrödinger-Poisson system

where \begin{document} $K(x)$ \end{document} is a positive and continuous potential and \begin{document} $f(u)$ \end{document} is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Under some suitable conditions, which are given in section 1, we prove that there exists some \begin{document} $\epsilon_{0}>0$ \end{document} such that for \begin{document} $0<\epsilon<\epsilon_{0}$ \end{document}, the above problem has infinitely many positive solutions by applying localized energy method. Our main result can be viewed as an extension to a recent result Theorem 1.1 of Ao and Wei in [3] and a result of Li, Peng and Wang in [26].

Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity
Xingxing Liu
2018, 38(11): 5505-5521 doi: 10.3934/dcds.2018242 +[Abstract](127) +[HTML](42) +[PDF](401.32KB)
Abstract:

In this paper, we consider the orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity, which admits the single peakons and multi-peakons. We firstly show the existence of the single peakon and prove two useful conservation laws. Then by constructing certain Lyapunov functionals, we give the proof of stability result of peakons in the energy space \begin{document}$ H^1(\mathbb{R})$\end{document}-norm.

Global existence for a two-component Camassa-Holm system with an arbitrary smooth function
Zeng Zhang and Zhaoyang Yin
2018, 38(11): 5523-5536 doi: 10.3934/dcds.2018243 +[Abstract](116) +[HTML](45) +[PDF](462.06KB)
Abstract:

This paper is concerned with a two-component integrable Camassa-Holm type system with arbitrary smooth function \begin{document}$ H$\end{document}. If the function $H$ belongs to a set \begin{document}$ \mathcal{H}$\end{document} (defined in Section 4), then we obtain the existence and uniqueness of global strong solutions and global weak solutions to the system. Our obtained results generalize and improve considerably recent results in [38,39].

Large data global regularity for the classical equivariant Skyrme model
Dan-Andrei Geba and Manoussos G. Grillakis
2018, 38(11): 5537-5576 doi: 10.3934/dcds.2018244 +[Abstract](88) +[HTML](36) +[PDF](606.05KB)
Abstract:

This article is concerned with the large data global regularity for the equivariant case of the classical Skyrme model and proves that this is valid for initial data in \begin{document}$H^s \times H^{s-1}(\mathbb{R}^3)$\end{document} with \begin{document}$s>7/2$\end{document}.

Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry
Gabriele Link
2018, 38(11): 5577-5613 doi: 10.3934/dcds.2018245 +[Abstract](103) +[HTML](41) +[PDF](630.08KB)
Abstract:

Let \begin{document}$X$\end{document} be a proper Hadamard space and \begin{document}$\Gamma <{\text{Is}}(X)$\end{document} a non-elementary discrete group of isometries with a rank one isometry. We discuss and prove Hopf-Tsuji-Sullivan dichotomy for the geodesic flow on the set of parametrized geodesics of the quotient \begin{document}$\Gamma \backslash X$\end{document} and with respect to Ricks' measure introduced in [35]. This generalizes previous work of the author and J. C. Picaud on Hopf-Tsuji-Sullivan dichotomy in the analogous manifold setting and with respect to Knieper's measure.

A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion
Perla El Kettani, Danielle Hilhorst and Kai Lee
2018, 38(11): 5615-5648 doi: 10.3934/dcds.2018246 +[Abstract](132) +[HTML](38) +[PDF](472.07KB)
Abstract:

In this paper, we prove a well posedness result for an initial boundary value problem for a stochastic nonlocal reaction-diffusion equation with nonlinear diffusion together with a nul-flux boundary condition in an open bounded domain of \begin{document} $\mathbb{R}^n$ \end{document} with a smooth boundary. We suppose that the additive noise is induced by a Q-Brownian motion.

Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation
Peng Gao
2018, 38(11): 5649-5684 doi: 10.3934/dcds.2018247 +[Abstract](136) +[HTML](51) +[PDF](572.52KB)
Abstract:

This work concerns the problem associated with averaging principle for a stochastic Kuramoto-Sivashinsky equation with slow and fast time-scales. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the stochastic Kuramoto-Sivashinsky equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single stochastic Kuramoto-Sivashinsky equation with a modified coefficient.

Error analysis of an ADI splitting scheme for the inhomogeneous Maxwell equations
Johannes Eilinghoff and Roland Schnaubelt
2018, 38(11): 5685-5709 doi: 10.3934/dcds.2018248 +[Abstract](107) +[HTML](46) +[PDF](519.8KB)
Abstract:

In this paper we investigate an alternating direction implicit (ADI) time integration scheme for the linear Maxwell equations with currents, charges and conductivity. We show its stability and efficiency. The main results establish that the scheme converges in a space similar to \begin{document}$H^{-1}$\end{document} with order two to the solution of the Maxwell system. Moreover, the divergence conditions in the system are preserved in \begin{document}$H^{-1}$\end{document} with order one.

Local correlation entropy
Vladimír Špitalský
2018, 38(11): 5711-5733 doi: 10.3934/dcds.2018249 +[Abstract](161) +[HTML](45) +[PDF](462.05KB)
Abstract:

Local correlation entropy, introduced by Takens in 1983, represents the exponential decay rate of the relative frequency of recurrences in the trajectory of a point, as the embedding dimension grows to infinity. In this paper we study relationship between the supremum of local correlation entropies and the topological entropy. For dynamical systems on topological graphs we prove that the two quantities coincide. Moreover, there is an uncountable set of points with local correlation entropy arbitrarily close to the topological entropy. On the other hand, we construct a strictly ergodic subshift with positive topological entropy having all local correlation entropies equal to zero. As a necessary tool, we derive an expected relationship between the local correlation entropies of a system and those of its iterates.

L1 semigroup generation for Fokker-Planck operators associated to general Lévy driven SDEs
Linghua Chen and Espen R. Jakobsen
2018, 38(11): 5735-5763 doi: 10.3934/dcds.2018250 +[Abstract](114) +[HTML](43) +[PDF](518.32KB)
Abstract:

We prove a new generation result in $L^1$ for a large class of non-local operators with non-degenerate local terms. This class contains the operators appearing in Fokker-Planck or Kolmogorov forward equations associated with Lévy driven SDEs, i.e. the adjoint operators of the infinitesimal generators of these SDEs. As a byproduct, we also obtain a new elliptic regularity result of independent interest. The main novelty in this paper is that we can consider very general Lévy operators, including state-space depending coefficients with linear growth and general Lévy measures which can be singular and have fat tails.

On dispersion decay for 3D Klein-Gordon equation
Elena Kopylova
2018, 38(11): 5765-5780 doi: 10.3934/dcds.2018251 +[Abstract](106) +[HTML](56) +[PDF](394.41KB)
Abstract:

We improve previous results on dispersion decay for 3D KleinGordon equation with generic potential. We develop a novel approach, which allows us to establish the decay in more strong norms and to weaken assumptions on the potential.

Minimum time problem with impulsive and ordinary controls
Monica Motta
2018, 38(11): 5781-5809 doi: 10.3934/dcds.2018252 +[Abstract](105) +[HTML](36) +[PDF](614.11KB)
Abstract:

Given a nonlinear control system depending on two controls \begin{document}$u$\end{document} and \begin{document}$v$\end{document}, with dynamics affine in the (unbounded) derivative of \begin{document}$u$\end{document} and a closed target set \begin{document}$\mathcal{S}$\end{document} depending both on the state and on the control \begin{document}$u$\end{document}, we study the minimum time problem with a bound on the total variation of \begin{document}$u$\end{document} and \begin{document}$u$\end{document} constrained in a closed, convex set \begin{document}$U$\end{document}, possibly with empty interior. We revisit several concepts of generalized control and solution considered in the literature and show that they all lead to the same minimum time function \begin{document}$T$\end{document}. Then we obtain sufficient conditions for the existence of an optimal generalized trajectory-control pair and study the possibility of Lavrentiev-type gap between the minimum time in the spaces of regular (that is, absolutely continuous) and generalized controls. Finally, under a convexity assumption on the dynamics, we characterize \begin{document}$T$\end{document} as the unique lower semicontinuous solution of a regular HJ equation with degenerate state constraints.

$L^p$ mapping properties for nonlocal Schrödinger operators with certain potentials
Woocheol Choi and Yong-Cheol Kim
2018, 38(11): 5811-5834 doi: 10.3934/dcds.2018253 +[Abstract](142) +[HTML](47) +[PDF](481.0KB)
Abstract:

In this paper, we consider nonlocal Schrödinger equations with certain potentials \begin{document}$V∈{\rm{RH}}^q$\end{document}(\begin{document}$q>\frac{n}{2s}>1$\end{document} and \begin{document}$0<s <1$\end{document}) of the form

where \begin{document}$L_K$\end{document} is an integro-differential operator. We denote the solution of the above equation by \begin{document}$\mathcal{S}_V f: = u$\end{document}, which is called the inverse of the nonlocal Schrödinger operator \begin{document}$L_K+V$\end{document} with potential \begin{document}$V$\end{document}; that is, \begin{document}$\mathcal{S}_V = (L_K+V)^{-1}$\end{document}. Then we obtain an improved version of the weak Harnack inequality of nonnegative weak subsolutions of the nonlocal equation

where \begin{document}$g∈ H^s(\mathbb{R}^n)$\end{document} and \begin{document}$\Omega$\end{document} is a bounded open domain in \begin{document}$\mathbb{R}^n$\end{document} with Lipschitz boundary, and also get an improved decay of a fundamental solution \begin{document}$\mathfrak{e}_V$\end{document} for \begin{document}$L_K+V$\end{document}. Moreover, we obtain \begin{document}$L^p$\end{document} and \begin{document}$L^p-L^q$\end{document} mapping properties of the inverse \begin{document}$\mathcal{S}_V$\end{document} of the nonlocal Schrödinger operator \begin{document}$L_K+V$\end{document}.

Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian
Vincenzo Ambrosio and Teresa Isernia
2018, 38(11): 5835-5881 doi: 10.3934/dcds.2018254 +[Abstract](164) +[HTML](43) +[PDF](716.51KB)
Abstract:

We consider a class of parametric Schrödinger equations driven by the fractional \begin{document}$p$\end{document}-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. Using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter.

Open maps: Small and large holes with unusual properties
Kevin G. Hare and Nikita Sidorov
2018, 38(11): 5883-5895 doi: 10.3934/dcds.2018255 +[Abstract](118) +[HTML](47) +[PDF](439.26KB)
Abstract:

Let X be a two-sided subshift on a finite alphabet endowed with a mixing probability measure which is positive on all cylinders in X. We show that there exists an arbitrarily small finite overlapping union of shifted cylinders which intersects every orbit under the shift map.

We also show that for any proper subshift Y of X there exists a finite overlapping unions of shifted cylinders such that its survivor set contains Y (in particular, it can have entropy arbitrarily close to the entropy of X). Both results may be seen as somewhat counter-intuitive.

Finally, we apply these results to a certain class of hyperbolic algebraic automorphisms of a torus.

Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow
Daehwan Kim and Juncheol Pyo
2018, 38(11): 5897-5919 doi: 10.3934/dcds.2018256 +[Abstract](131) +[HTML](42) +[PDF](2554.13KB)
Abstract:

Translating soliton is a special solution for the mean curvature flow (MCF) and the parabolic rescaling model of type Ⅱ singularities for the MCF. By introducing an appropriate coordinate transformation, we first show that there exist complete helicoidal translating solitons for the MCF in \begin{document}$\mathbb{R}^{3}$\end{document} and we classify the profile curves and analyze their asymptotic behavior. We rediscover the helicoidal translating solitons for the MCF which are founded by Halldorsson [10]. Second, for the pinch zero we rediscover rotationally symmetric translating solitons in \begin{document}$\Bbb R^{n+1}$\end{document} and analyze the asymptotic behavior of the profile curves using a dynamical system. Clearly rotational hypersurfaces are foliated by spheres. We finally show that translating solitons foliated by spheres become rotationally symmetric translating solitons with the axis of revolution parallel to the translating direction. Hence, we obtain that any translating soliton foliated by spheres becomes either an n-dimensional translating paraboloid or a winglike translator.

The diffusion phenomenon for damped wave equations with space-time dependent coefficients
Montgomery Taylor
2018, 38(11): 5921-5941 doi: 10.3934/dcds.2018257 +[Abstract](139) +[HTML](56) +[PDF](491.59KB)
Abstract:

We introduce a method to study the long-time behavior of solutions to damped wave equations, where the coefficients of the equations are space-time dependent. We show that solutions exhibit the diffusion phenomenon, connecting their asymptotic behaviors with the asymptotic behaviors of solutions to corresponding parabolic equations. Sharp decay estimates for solutions to damped wave equations are given, and decay estimates for derivatives of solutions are also discussed.

Global existence and boundedness in a chemorepulsion system with superlinear diffusion
Marcel Freitag
2018, 38(11): 5943-5961 doi: 10.3934/dcds.2018258 +[Abstract](160) +[HTML](48) +[PDF](482.09KB)
Abstract:

In a bounded domain \begin{document}$\Omega\subset\mathbb{R}^n$\end{document}, where \begin{document}$n\ge 3$\end{document}, we consider the quasilinear parabolic-parabolic Keller-Segel system

with homogeneous Neumann boundary conditions. We will find that the condition \begin{document}$D(u)\geq Cu^{m-1}$\end{document} suffices to prove the uniqueness and global existence of solutions along with their boundedness if \begin{document}$D(0)>0$\end{document} and \begin{document}$m>1+\frac{(n-2)(n-1)}{n^2}$\end{document} which is a very different result from what we know about the same system with nonnegative sensitivity. In the case of degenerate diffusion (\begin{document}$D(0) = 0$\end{document}) and for the same parameters, locally bounded global weak solutions will be established.

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