# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

August 2017 , Volume 37 , Issue 7

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2017, 37(7): 3521-3530 doi: 10.3934/dcds.2017150 +[Abstract](364) +[HTML](3) +[PDF](279.5KB)
Abstract:

We consider the semilinear elliptic equation \begin{document}$-\Delta u =\lambda f(u)$\end{document} in a smooth bounded domain \begin{document}$\Omega$\end{document} of \begin{document}$\Bbb{R}^{n}$\end{document} with Dirichlet boundary condition, where \begin{document}$f$\end{document} is a \begin{document}$C^{1}$\end{document} positive and nondeccreasing function in \begin{document}$[0, \infty)$\end{document} such that \begin{document}$\frac{f(t)}{t} \rightarrow \infty$\end{document} as \begin{document}$t \rightarrow \infty$\end{document}. When \begin{document}$\Omega$\end{document} is an arbitrary domain and \begin{document}$f$\end{document} is not necessarily convex, the boundedness of the extremal solution \begin{document}$u^{*}$\end{document} is known only for \begin{document}$n = 2$\end{document}, established by X. Cabré[5]. In this paper, we prove this for higher dimensions depending on the nonlinearity \begin{document}$f$\end{document}. In particular, we prove that if

where \begin{document}$F(t)=\int_{0}^{t}f(s)ds$\end{document}, then \begin{document}$u^{*} \in L^{\infty}(\Omega)$\end{document}, for \begin{document}$n \leq 6$\end{document}. Also, if \begin{document}$\beta_{-}=\beta_{+}>\frac{1}{2}$\end{document} or \begin{document}$\frac{1}{2} < \beta_{-}\leq \beta_{+} < \frac{7}{10}$\end{document}, then \begin{document}$u^{*} \in L^{\infty}(\Omega)$\end{document}, for \begin{document}$n \leq 9$\end{document}. Moreover, under the sole condition that \begin{document}$\beta_{-} > \frac{1}{2}$\end{document} we have \begin{document}$u^{*} \in H^{1}_{0}(\Omega)$\end{document} for \begin{document}$n \geq 1$\end{document}. The same is true if for some \begin{document}$\epsilon > 0$\end{document} we have

which improves a similar result by Brezis and Vázquez [4].

2017, 37(7): 3531-3544 +[Abstract](569) +[HTML](23) +[PDF](370.4KB)
Abstract:

We combine the classical Gromov-Hausdorff metric [5] with the \begin{document}$C^0$\end{document} distance to obtain the \begin{document}$C^0$\end{document}-Gromov-Hausdorff distance between maps of possibly different metric spaces. The latter is then combined with Walters's topological stability [11] to obtain the notion of topologically GH-stable homeomorphism. We prove that there are topologically stable homeomorphism which are not topologically GH-stable. Also that every topological GH-stable circle homeomorphism is topologically stable. Afterwards, we prove that every expansive homeomorphism with the pseudo-orbit tracing property of a compact metric space is topologically GH-stable. This is related to Walters's stability theorem [11]. Finally, we extend the topological GH-stability to continuous maps and prove the constant maps on compact homogeneous manifolds are topologically GH-stable.

2017, 37(7): 3545-3566 doi: 10.3934/dcds.2017152 +[Abstract](583) +[HTML](2) +[PDF](576.4KB)
Abstract:

Abstract. In this paper, the equivariant degree theory is used to analyze the occurrence of the Hopf bifurcation under effectively verifiable mild conditions. We combine the abstract result with standard interval polynomial techniques based on Kharitonov's theorem to show the existence of a branch of periodic solutions emanating from the equilibrium in the settings relevant to robust control. The results are illustrated with a number of examples.

2017, 37(7): 3567-3585 doi: 10.3934/dcds.2017153 +[Abstract](599) +[HTML](7) +[PDF](6022.0KB)
Abstract:

The goal of this paper is to study the family of singular perturbations of Blaschke products given by \begin{document} $B_{a, λ}(z)=z^3\frac{z-a}{1-\overline{a}z}+\frac{λ}{z^2}$ \end{document}. We focus on the study of these rational maps for parameters \begin{document} $a$ \end{document} in the punctured disk \begin{document} $\mathbb{D}^*$ \end{document} and \begin{document} $|λ|$ \end{document} small. We prove that, under certain conditions, all Fatou components of a singularly perturbed Blaschke product \begin{document} $B_{a, λ}$ \end{document} have finite connectivity but there are components of arbitrarily large connectivity within its dynamical plane. Under the same conditions we prove that the Julia set is the union of countably many Cantor sets of quasicircles and uncountably many point components.

2017, 37(7): 3587-3599 doi: 10.3934/dcds.2017154 +[Abstract](519) +[HTML](13) +[PDF](412.5KB)
Abstract:

Let \begin{document} $0 < \alpha < 2$ \end{document} be any real number and let \begin{document} $\Omega$ \end{document} be an open domain in \begin{document} $\mathbb R^{n}$ \end{document}. Consider the following Dirichlet problem of a semi-linear equation involving the fractional Laplacian:

\begin{document}$$$\left\{\begin{array}{ll}(-\Delta)^{\alpha/2} u(x)=f(x,u,\nabla{u}),~u(x)>0,&\qquad x\in{\Omega}, \\u(x)\equiv0,&\qquad x\notin{\Omega}.\end{array}\right. \tag{1}\label{p1}$$$\end{document}

In this paper, instead of using the conventional extension method introduced by Caffarelli and Silvestre, we employ a direct method of moving planes for the fractional Laplacian to obtain the monotonicity and symmetry of the positive solutions of a semi-linear equation involving the fractional Laplacian. By using the integral definition of the fractional Laplacian, we first introduce various maximum principles which play an important role in the process of moving planes. Then we establish the monotonicity and symmetry of positive solutions of the semi-linear equations involving the fractional Laplacian.

2017, 37(7): 3601-3623 doi: 10.3934/dcds.2017155 +[Abstract](431) +[HTML](3) +[PDF](510.1KB)
Abstract:

In this paper we consider a definition of Morse-Smale evolution process that extends the notion of Morse-Smale dynamical system to the nonautonomous framework. In particular we consider nonautonomous perturbations of autonomous systems. In this case our definition of Morse-Smale evolution process holds for perturbations of Morse-Smale autonomous systems with or without periodic orbits. We establish that small nonautonomous perturbations of autonomous Morse-Smale evolution processes derived from certain nonautonomous differential equations are Morse-Smale evolution processes. We apply our results to examples of scalar parabolic semilinear differential equations generating evolution processes and possessing periodic orbits.

2017, 37(7): 3625-3699 doi: 10.3934/dcds.2017156 +[Abstract](402) +[HTML](12) +[PDF](919.3KB)
Abstract:

We prove locally in time the existence of the unique smooth solution (including smooth interface) to the multidimensional free boundary problem for the thin film equation with the mobility n = 2 in the case of partial wetting. We also obtain the Schauder estimates and solvability for the Dirichlet and the Neumann problem for a linear degenerate parabolic equation of fourth order.

2017, 37(7): 3701-3719 doi: 10.3934/dcds.2017157 +[Abstract](284) +[HTML](0) +[PDF](412.7KB)
Abstract:

We study k-bonacci substitutions through the point of view of thermodynamic formalism. For each substitution we define a renormalization operator associated to it and examine its iterates over potentials in a certain class. We also study the pressure function associated to potentials in this class and prove the existence of a freezing phase transition which is realized by the only ergodic measure on the subshift associated to the substitution.

2017, 37(7): 3721-3747 doi: 10.3934/dcds.2017158 +[Abstract](349) +[HTML](1) +[PDF](1651.5KB)
Abstract:

The quadratic nonlinear wave equation on a one-dimensional torus with small initial values located in a single Fourier mode is considered. In this situation, the formation of metastable energy strata has recently been described and their long-time stability has been shown. The topic of the present paper is the correct reproduction of these metastable energy strata by a numerical method. For symplectic trigonometric integrators applied to the equation, it is shown that these energy strata are reproduced even on long time intervals in a qualitatively correct way.

2017, 37(7): 3749-3786 doi: 10.3934/dcds.2017159 +[Abstract](478) +[HTML](8) +[PDF](711.8KB)
Abstract:

The paper is concerned with a system of two coupled time-independent Gross-Pitaevskii equations in \begin{document} $\mathbb{R}^2$ \end{document}, which is used to model two-component Bose-Einstein condensates with both attractive intraspecies and attractive interspecies interactions. This system is essentially an eigenvalue problem of a stationary nonlinear Schrödinger system in \begin{document} $\mathbb{R}^2$ \end{document}, and solutions of the problem are obtained by seeking minimizers of the associated variational functional with constrained mass (i.e. \begin{document} $L^2-$ \end{document}norm constaints). Under a certain type of trapping potentials \begin{document} $V_i(x)$ \end{document} (\begin{document} $i=1, 2$ \end{document}), the existence, non-existence and uniqueness of this kind of solutions are studied. Moreover, by establishing some delicate energy estimates, we show that each component of the solutions blows up at the same point (i.e., one of the global minima of \begin{document} $V_i(x)$ \end{document}) when the total interaction strength of intraspecies and interspecies goes to a critical value. An optimal blowing up rate for the solutions of the system is also given.

2017, 37(7): 3787-3804 doi: 10.3934/dcds.2017160 +[Abstract](354) +[HTML](12) +[PDF](436.7KB)
Abstract:

In this paper, we formulate a conjecture for a measure-preservation criterion of 1-Lipschitz functions defined on the ring Zp of p-adic integers, in terms of Mahler's expansion. We then provide evidence for this conjecture in the case that p = 3, and verify that it also holds for a wider class of 1-Lipschitz functions that are everywhere differentiable on Zp, which we call $\mathcal{ B}$-functions, in the sense of Anashin.

2017, 37(7): 3805-3830 doi: 10.3934/dcds.2017161 +[Abstract](237) +[HTML](2) +[PDF](523.6KB)
Abstract:

Let \begin{document}$M$\end{document} be a model set meeting two simple conditions: (1) the internal group \begin{document}$H$\end{document} is \begin{document}$\mathbb{R}^n$\end{document} (or a product of \begin{document}$\mathbb{R}^n$\end{document} and a finite group) and (2) the window \begin{document}$W$\end{document} is a finite union of disjoint polyhedra. Then any Delone set with finite local complexity (FLC) that is topologically conjugate to \begin{document}$M$\end{document} is mutually locally derivable (MLD) to a model set \begin{document}$M'$\end{document} that has the same internal group and window as \begin{document}$M$\end{document}, but has a different projection from \begin{document}$H × \mathbb{R}^d$\end{document} to \begin{document}$\mathbb{R}^d$\end{document}. In cohomological terms, this means that the group \begin{document}$H^1_{an}(M, \mathbb{R})$\end{document} of asymptotically negligible classes has dimension \begin{document}$n$\end{document}. We also exhibit a counterexample when the second hypothesis is removed, constructing two topologically conjugate FLC Delone sets, one a model set and the other not even a Meyer set.

2017, 37(7): 3831-3866 doi: 10.3934/dcds.2017162 +[Abstract](450) +[HTML](7) +[PDF](651.7KB)
Abstract:

We consider the defocusing energy-critical nonlinear Schrödinger equation with inverse-square potential \begin{document}$iu_t = -Δ u + a|x|^{-2}u + |u|^4u$\end{document} in three space dimensions. We prove global well-posedness and scattering for \begin{document}$a > - \frac{1}{4} + \frac{1}{{25}}$\end{document}. We also carry out the variational analysis needed to treat the focusing case.

2017, 37(7): 3867-3903 doi: 10.3934/dcds.2017163 +[Abstract](296) +[HTML](0) +[PDF](633.5KB)
Abstract:

We give a fairly complete characterization of the exact components of a large class of uniformly expanding Markov maps of \begin{document}${\mathbb{R}}$\end{document}. Using this result, for a class of \begin{document}\end{document} $\mathbb{Z}$-invariant maps and finite modifications thereof, we prove certain properties of infinite mixing recently introduced by the author.

2017, 37(7): 3905-3920 doi: 10.3934/dcds.2017164 +[Abstract](320) +[HTML](6) +[PDF](410.8KB)
Abstract:

In this paper, we study the existence of SRB measures for infinite dimensional dynamical systems in a Banach space. We show that if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions, then it has an SRB measure.

2017, 37(7): 3921-3938 doi: 10.3934/dcds.2017165 +[Abstract](391) +[HTML](2) +[PDF](444.3KB)
Abstract:

This paper studies the local existence of strong solutions to the Cauchy problem of the 2D simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows, coupled via $ρ$ (the density of the fluid), $u$ (the velocity of the field), and $d$ (the macroscopic/continuum molecular orientations). Notice that the technique used for the corresponding 3D local well-posedness of strong solutions fails treating the 2D case, because the $L^p$-norm ($p>2$) of the velocity $u$ cannot be controlled in terms only of $ρ^{\frac{1}{2}}u$ and $\nabla u$ here. In the present paper, under the framework of weighted approximation estimates introduced in [J. Li, Z. Liang, On classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl. (2014) 640-671] for Navier-Stokes equations, we obtain the local existence of strong solutions to the 2D compressible nematic liquid crystal flows.

2017, 37(7): 3939-3961 doi: 10.3934/dcds.2017167 +[Abstract](468) +[HTML](2) +[PDF](464.9KB)
Abstract:

A type of nonautonomous n-dimensional state-dependent delay differential equation (SDDE) is studied. The evolution law is supposed to satisfy standard conditions ensuring that it can be imbedded, via the Bebutov hull construction, in a new map which determines a family of SDDEs when it is evaluated along the orbits of a flow on a compact metric space. Additional conditions on the initial equation, inherited by those of the family, ensure the existence and uniqueness of the maximal solution of each initial value problem. The solutions give rise to a skew-product semiflow which may be noncontinuous, but which satisfies strong continuity properties. In addition, the solutions of the variational equation associated to the SDDE determine the Fréchet differential with respect to the initial state of the orbits of the semiflow at the compatibility points. These results are key points to start using topological tools in the analysis of the long-term behavior of the solution of this type of nonautonomous SDDEs.

2017, 37(7): 3963-3987 doi: 10.3934/dcds.2017168 +[Abstract](584) +[HTML](6) +[PDF](522.6KB)
Abstract:

In this paper, we study a class of nonlinear Schrödinger equations involving the fractional Laplacian and the nonlinearity term with critical Sobolev exponent. We assume that the potential of the equations includes a parameter $λ$. Moreover, the potential behaves like a potential well when the parameter λ is large. Using variational methods, combining Nehari methods, we prove that the equation has a least energy solution which, as the parameter λ large, localizes near the bottom of the potential well. Moreover, if the zero set int \begin{document} $V^{-1}(0)$ \end{document} of \begin{document} $V(x)$ \end{document} includes more than one isolated component, then \begin{document} $u_\lambda (x)$ \end{document} will be trapped around all the isolated components. However, in Laplacian case when \begin{document} $s=1$ \end{document}, for \begin{document} $\lambda$ \end{document} large, the corresponding least energy solution will be trapped around only one isolated component and will become arbitrary small in other components of int \begin{document} $V^{-1}(0)$ \end{document}. This is the essential difference with the Laplacian problems since the operator \begin{document} $(-Δ)^{s}$ \end{document} is nonlocal.

2017, 37(7): 3989-4018 doi: 10.3934/dcds.2017169 +[Abstract](467) +[HTML](1) +[PDF](362.2KB)
Abstract:

We prove new variational properties of the spatial isosceles orbits in the equal-mass three-body problem and analyze their linear stabilities in both the full phase space \begin{document}$\mathbb{R}^{12}$\end{document} and a symmetric subspace Γ. We prove that each spatial isosceles orbit is an action minimizer of a two-point free boundary value problem with non-symmetric boundary settings. The spatial isosceles orbits form a one-parameter set with rotation angle θ as the parameter. This set of orbits always lies in a symmetric subspace Γ and we show that their linear stabilities in the full phase space \begin{document}$\mathbb{R}^{12}$\end{document} can be simplified to two separated sub-problems: linear stabilities in Γ and \begin{document}$(\mathbb{R}^{12} \setminus Γ) \cup \{0\}$\end{document}. By applying Roberts' symmetry reduction method, we prove that the orbits are always unstable in the full phase space \begin{document}$\mathbb{R}^{12}$\end{document}, but it is linearly stable in Γ when \begin{document}$θ ∈ [0.33π, 0.48 π] \cup [0.52 π, 0.78 π]$\end{document}.

2017, 37(7): 4019-4034 doi: 10.3934/dcds.2017170 +[Abstract](292) +[HTML](4) +[PDF](436.6KB)
Abstract:

We study parametrised families of piecewise expanding interval mappings \begin{document}$T_a \colon [0,1] \to [0,1]$\end{document} with absolutely continuous invariant measures \begin{document}$\mu_a$\end{document} and give sufficient conditions for a point \begin{document}$X(a)$\end{document} to be typical with respect to \begin{document}$(T_a, \mu_a)$\end{document} for almost all parameters a. This is similar to a result by D.Schnellmann, but with different assumptions.

2017, 37(7): 4035-4051 doi: 10.3934/dcds.2017171 +[Abstract](406) +[HTML](3) +[PDF](504.3KB)
Abstract:

In this paper, we study an anomalous diffusion model of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we are concerned with the following initial-boundary value problem involving the fractional $p$-Laplacian $\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+M([u]_{s, p}^{p}\text{)}(-\Delta)_{p}^{s}u=f(x, t) & \text{in }\Omega \times {{\mathbb{R}}^{+}}, {{\partial }_{t}}u=\partial u/\partial t, \\ u(x, 0)={{u}_{0}}(x) & \text{in }\Omega, \\ u=0\ & \text{in }{{\mathbb{R}}^{N}}\backslash \Omega, \\\end{array}\text{ }\ \ \right.$ where $[u]_{s, p}$ is the Gagliardo $p$-seminorm of $u$, $Ω\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partialΩ$, $1 < p < N/s$, with $0 < s < 1$, the main Kirchhoff function $M:\mathbb{R}^{ + }_{0} \to \mathbb{R}^{ + }$ is a continuous and nondecreasing function, $(-Δ)_p^s$ is the fractional $p$-Laplacian, $u_0$ is in $L^2(Ω)$ and $f∈ L^2_{\rm loc}(\mathbb{R}^{ + }_0;L^2(Ω))$. Under some appropriate conditions, the well-posedness of solutions for the problem above is studied by employing the sub-differential approach. Finally, the large-time behavior and extinction of solutions are also investigated.

2017, 37(7): 4053-4069 doi: 10.3934/dcds.2017172 +[Abstract](413) +[HTML](1) +[PDF](424.4KB)
Abstract:

This paper deals with a characterization of asymptotic stability for a class of dynamical systems in terms of smooth Lyapunov pairs. We point out that well known converse Lyapunov results for differential inclusions cannot be applied to this class of dynamical systems. Following an abstract approach we put an assumption on the trajectories of the dynamical systems which demands for an estimate of the difference between trajectories. Under this assumption, we prove the existence of a $C^∞$-smooth Lyapunov pair. We also show that this assumption is satisfied by differential inclusions defined by Lipschitz continuous set-valued maps taking nonempty, compact and convex values.

2017, 37(7): 4071-4089 doi: 10.3934/dcds.2017173 +[Abstract](489) +[HTML](1) +[PDF](514.5KB)
Abstract:

The original proof of Dacorogna-Moser theorem on the prescribed Jacobian PDE, \begin{document}$\text{det}\, \nabla\varphi=f$\end{document} , can be modified in order to obtain control of support of the solutions from that of the initial data, while keeping optimal regularity. Briefly, under the usual conditions, a solution diffeomorphism \begin{document}$\varphi$\end{document} satisfying \begin{document}$\text{supp}(f-1)\subset\varOmega\Longrightarrow\text{supp}(\varphi-\text{id})\subset\varOmega$\end{document} can be found and \begin{document}$\varphi$\end{document} is still of class $C^{r+1, α}$ if $f$ is $C^{r, α}$, the domain of $f$ being a bounded connected open \begin{document}$C^{r+2, α}$ $\end{document} set \begin{document}$\varOmega\subset\mathbb{R}^{n}$\end{document} . 2017, 37(7): 4091-4108 doi: 10.3934/dcds.2017174 +[Abstract](450) +[HTML](4) +[PDF](450.4KB) Abstract: In this article we investigate the nonlinear stability of Hasimoto solitons, in energy space, for a fourth order Schrödinger equation (4NLS) which arises in the context of the vortex filament. The proof relies on a suitable Lyapunov functional, at the \begin{document}$H^2$\end{document} level, which allows us to describe the dynamics of small perturbations. This stability result is also extended to Sobolev spaces \begin{document}$H^m$\end{document} for all \begin{document}$m∈\mathbb{Z}_+$\end{document} by employing the infinite conservation laws of 4NLS. 2017, 37(7): 4109-4129 doi: 10.3934/dcds.2017175 +[Abstract](393) +[HTML](11) +[PDF](436.6KB) Abstract: We study nonlinear elliptic equations of strong$p(x)$-Laplacian type to obtain an interior Calderón-Zygmund type estimates by finding a correct regularity assumption on the variable exponent$p(x)\$. Our proof is based on the maximal function technique and the appropriate localization method.

2017, 37(7): 4131-4158 doi: 10.3934/dcds.2017176 +[Abstract](374) +[HTML](12) +[PDF](1158.2KB)
Abstract:

We analyze the effect of Robin boundary conditions in a mathematical model for a mitochondria swelling in a living organism. This is a coupled PDE/ODE model for the dependent variables calcium ion contration and three fractions of mitochondria that are distinguished by their state of swelling activity. The model assumes that the boundary is a permeable 'membrane', through which calcium ions can both enter or leave the cell. Under biologically relevant assumptions on the data, we prove the well-posedness of solutions of the model and study the asymptotic behavior of its solutions. We augment the analysis of the model with computer simulations that illustrate the theoretically obtained results.

2017  Impact Factor: 1.179