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## Discrete & Continuous Dynamical Systems - A

July 2016 , Volume 36 , Issue 7

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2016, 36(7): 3483-3510
doi: 10.3934/dcds.2016.36.3483

*+*[Abstract](2148)*+*[PDF](550.3KB)**Abstract:**

We prove existence and conditional energetic stability of solitary-wave solutions for the two classes of pseudodifferential equations \begin{equation*} u_t+\left(f(u)\right)_x-\left(L u\right)_x=0 \end{equation*} and \begin{equation*} u_t+\left(f(u)\right)_x+\left(L u\right)_t=0, \end{equation*} where $f$ is a nonlinear term, typically of the form $c|u|^p$ or $cu|u|^{p-1}$, and $L$ is a Fourier multiplier operator of positive order. The former class includes for instance the Whitham equation with capillary effects and the generalized Korteweg-de Vries equation, and the latter the Benjamin-Bona-Mahony equation. Existence and conditional energetic stability results have earlier been established using the method of concentration-compactness for a class of operators with symbol of order $s\geq 1$. We extend these results to symbols of order $0 < s < 1$, thereby improving upon the results for general operators with symbol of order $s\geq 1$ by enlarging both the class of linear operators and nonlinearities admitting existence of solitary waves. Instead of using abstract operator theory, the new results are obtained by direct calculations involving the nonlocal operator $L$, something that gives us the bounds and estimates needed for the method of concentration-compactness.

2016, 36(7): 3511-3518
doi: 10.3934/dcds.2016.36.3511

*+*[Abstract](1908)*+*[PDF](321.7KB)**Abstract:**

We prove that the genus two surface admits a cw-expansive homeomorphism with a fixed point whose local stable set is not locally connected.

2016, 36(7): 3519-3543
doi: 10.3934/dcds.2016.36.3519

*+*[Abstract](2352)*+*[PDF](451.6KB)**Abstract:**

We obtain sufficient conditions for the differentiability of solutions to stationary Fokker--Planck--Kolmogorov equations with respect to a parameter. In particular, this gives conditions for the differentiability of stationary distributions of diffusion processes with respect to a parameter.

2016, 36(7): 3545-3601
doi: 10.3934/dcds.2016.36.3545

*+*[Abstract](2327)*+*[PDF](959.0KB)**Abstract:**

In this paper we study the Sotomayor-Teixeira regularization of a general visible fold singularity of a planar Filippov system. Extending Geometric Fenichel Theory beyond the fold with asymptotic methods, we determine the deviation of the orbits of the regularized system from the generalized solutions of the Filippov one. This result is applied to the regularization of global sliding bifurcations as the Grazing-Sliding of periodic orbits and the Sliding Homoclinic to a Saddle, as well as to some classical problems in dry friction.

Roughly speaking, we see that locally, and also globally, the regularization of the bifurcations preserve the topological features of the sliding ones.

2016, 36(7): 3603-3621
doi: 10.3934/dcds.2016.36.3603

*+*[Abstract](1519)*+*[PDF](486.4KB)**Abstract:**

Let $\Omega_\ell = \ell\omega_1 \times \omega_2$ where $\omega_1 \subset \mathbb{R}^p$ and $\omega_2 \subset \mathbb{R}^{n-p}$ are assumed to be open and bounded. We consider the following minimization problem: $$E_{\Omega_\ell}(u_\ell) = \min_{u\in W_0^{1,q}(\Omega_\ell)}E_{\Omega_\ell}(u)$$ where $E_{\Omega_\ell}(u) = \int_{\Omega_\ell}F(\nabla u)-fu$, $F$ is a convex function and $f\in L^{q'}(\omega_2)$. We are interested in studying the asymptotic behavior of the solution $u_\ell$ as $\ell$ tends to infinity.

2016, 36(7): 3623-3638
doi: 10.3934/dcds.2016.36.3623

*+*[Abstract](2215)*+*[PDF](419.4KB)**Abstract:**

A vector field has the periodic shadowing property if for any $\varepsilon>0$ there is $d>0$ such that, for any periodic $d$-pseudo orbit $g$ there exists a periodic orbit or a singularity in which $g$ is $\varepsilon$-shadowed. In this paper, we show that a vector field is in the $C^1$ interior of the set of vector fields satisfying the periodic shadowing property if and only if it is $\Omega$-stable. More precisely, we prove that the $C^1$ interior of the set of vector fields satisfying the orbital periodic shadowing property is a subset of the set of $\Omega$-stable vector fields.

2016, 36(7): 3639-3650
doi: 10.3934/dcds.2016.36.3639

*+*[Abstract](2512)*+*[PDF](453.0KB)**Abstract:**

The blowup is studied for the nonlinear Schrödinger equation $iu_{t}+\Delta u+ |u|^{p-1}u=0$ with $p$ is odd and $p\ge 1+\frac 4{N-2}$ (the energy-critical or energy-supercritical case). It is shown that the solution with negative energy $E(u_0)<0$ blows up in finite or infinite time. A new proof is also presented for the previous result in [9], in which a similar result in a case of energy-subcritical was shown.

2016, 36(7): 3651-3675
doi: 10.3934/dcds.2016.36.3651

*+*[Abstract](1738)*+*[PDF](552.7KB)**Abstract:**

Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of existence of a conservative renormalization fixed point.

Furthermore, it has been shown by Gaidashev, Johnson and Martens that

*infinitely renormalizable maps*in a neighborhood of this fixed point admit invariant Cantor sets with vanishing Lyapunov exponents on which dynamics for any two maps is smoothly conjugate.

This rigidity is a consequence of an interplay between the decay of geometry and the convergence rate of renormalization towards the fixed point.

In this paper we prove a result which is crucial for a demonstration of rigidity: that an upper bound on this convergence rate of renormalizations of infinitely renormalizable maps is sufficiently small.

2016, 36(7): 3677-3703
doi: 10.3934/dcds.2016.36.3677

*+*[Abstract](2898)*+*[PDF](565.7KB)**Abstract:**

In this note we describe how the Neumann homogenization of fully nonlinear elliptic equations can be recast as the study of nonlocal (integro-differential) equations involving elliptic integro-differential operators on the boundary. This is motivated by a new integro-differential representation for nonlinear operators with a comparison principle which we also introduce. In the simple case that the original domain is an infinite strip with almost periodic Neumann data, this leads to an almost periodic homogenization problem involving a fully nonlinear integro-differential operator on the Neumann boundary. This method gives a new proof-- which was left as an open question in the earlier work of Barles- Da Lio- Lions- Souganidis (2008)-- of the result obtained recently by Choi-Kim-Lee (2013), and we anticipate that it will generalize to other contexts.

2016, 36(7): 3705-3717
doi: 10.3934/dcds.2016.36.3705

*+*[Abstract](1981)*+*[PDF](215.5KB)**Abstract:**

The commonly used topological entropy $h_{top}(\mathcal{U})$ of the multi-dimensional shift space $\mathcal{U}$ is the rectangular spatial entropy $h_{r}(\mathcal{U})$ which is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space $\mathbb{Z}^{d}$, $d\geq 2$. This work studies spatial entropy $h_{\Omega}(\mathcal{U})$ of shift space $\mathcal{U}$ on general expanding system $\Omega=\{\Omega(n)\}_{n=1}^{\infty}$ where $\Omega(n)$ is increasing finite sublattices and expands to $\mathbb{Z}^{d}$. $\Omega$ is called genuinely $d$-dimensional if $\Omega(n)$ contains no lower-dimensional part whose size is comparable to that of its $d$-dimensional part. We show that $h_{r}(\mathcal{U})$ is the supremum of $h_{\Omega}(\mathcal{U})$ for all genuinely $d$-dimensional $\Omega$. Furthermore, when $\Omega$ is genuinely $d$-dimensional and satisfies certain conditions, then $h_{\Omega}(\mathcal{U})=h_{r}(\mathcal{U})$. On the contrary, when $\Omega(n)$ contains a lower-dimensional part which is comparable to its $d$-dimensional part, then $h_{r}(\mathcal{U}) < h_{\Omega}(\mathcal{U})$ for some $\mathcal{U}$. Therefore, $h_{r}(\mathcal{U})$ is appropriate to be the $d$-dimensional spatial entropy.

2016, 36(7): 3719-3739
doi: 10.3934/dcds.2016.36.3719

*+*[Abstract](2212)*+*[PDF](466.8KB)**Abstract:**

We study the interior approximate controllability of fractional wave equations with the fractional Caputo derivative associated with a non-negative self-adjoint operator satisfying the unique continuation property. Some well-posedness and fine regularity properties of solutions to fractional wave and fractional backward wave type equations are also obtained. As an example of applications of our results we obtain that if $1<\alpha<2$ and $\Omega\subset\mathbb{R}^N$ is a smooth connected open set with boundary $\partial\Omega$, then the system $\mathbb D_t^\alpha u+A_Bu=f$ in $\Omega\times (0,T)$, $u(\cdot,0)=u_0$, $\partial_tu(\cdot,0)=u_1$, is approximately controllable for any $T>0$, $(u_0,u_1)\in V_{\frac{1}{\alpha}}\times L^2(\Omega)$, $\omega\subset\Omega$ any open set and any $f\in C_0^\infty(\omega\times (0,T))$. Here, $A_B$ can be the realization in $L^2(\Omega)$ of a symmetric non-negative uniformly elliptic operator with Dirichlet or Robin boundary conditions, or the realization in $L^2(\Omega)$ of the fractional Laplace operator $(-\Delta)^s$ ($0< s <1$) with the Dirichlet boundary condition ($u=0$ on $\mathbb{R}^N\setminus\Omega$) and the space $V_{\frac{1}{\alpha}}$ denotes the domain of the fractional power of order $\frac{1}{\alpha}$ of the operator $A_B$.

2016, 36(7): 3741-3774
doi: 10.3934/dcds.2016.36.3741

*+*[Abstract](2027)*+*[PDF](553.2KB)**Abstract:**

In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the $L_p$ in time and the $L_q$ in space framework with $2< p <\infty$ and $N< q <\infty$ under the assumption that the initial domain is a uniform $W^{2-1/q}_q$ domain in $\mathbb{R}^N (N\ge 2)$. After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal $L_p$-$L_q$ regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of $\mathcal{R}$-bounded solution operator to resolvent problem corresponding to linearized problem. The $\mathcal{R}$-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal $L_p$-$L_q$ regularity theorem.

A Schechter type critical point result in annular conical domains of a Banach space and applications

2016, 36(7): 3775-3789
doi: 10.3934/dcds.2016.36.3775

*+*[Abstract](1710)*+*[PDF](397.1KB)**Abstract:**

Using Ekeland's variational principle we obtain a critical point theorem of Schechter type for extrema of a functional in an annular conical domain of a Banach space. The result can be seen as a variational analogue of Krasnoselskii's fixed point theorem in cones and can be applied for the existence, localization and multiplicity of the positive solutions of variational problems. The result is then applied to $p$-Laplace equations, where the geometric condition on the boundary of the annular conical domain is established via a weak Harnack type inequality given in terms of the energetic norm. This method can be applied also to other homogeneous operators in order to obtain existence, multiplicity or infinitely many solutions for certain classes of quasilinear equations.

2016, 36(7): 3791-3810
doi: 10.3934/dcds.2016.36.3791

*+*[Abstract](1922)*+*[PDF](444.7KB)**Abstract:**

This paper is concerned with the properties of solutions for the weighted Hardy-Littlewood-Sobolev type integral system \begin{equation} \left \{ \begin{array}{l} u(x) = \frac{1}{|x|^{\alpha}}\int_{R^{n}} \frac{v^q(y)}{|y|^{\beta}|x-y|^{\lambda}} dy,\\ v(x) = \frac{1}{|x|^{\beta}}\int_{R^{n}} \frac{u^p(y)}{|y|^{\alpha}|x-y|^{\lambda}} dy \end{array} \right. （1） \end{equation} and the fractional order partial differential system \begin{equation} \label{PDE} \left\{\begin{array}{ll} (-\Delta)^{\frac{n-\lambda}{2}}(|x|^{\alpha}u(x)) =|x|^{-\beta} v^{q}(x), \\ (-\Delta)^{\frac{n-\lambda}{2}}(|x|^{\beta}v(x)) =|x|^{-\alpha} u^p(x). \end{array} （2） \right. \end{equation} Here $x \in R^n \setminus \{0\}$. Due to $0 < p, q < \infty$, we need more complicated analytical techniques to handle the case $0< p <1$ or $0< q <1$. We first establish the equivalence of integral system (1) and fractional order partial differential system (2). For integral system (1), we prove that the integrable solutions are locally bounded. In addition, we also show that the positive locally bounded solutions are symmetric and decreasing about some axis by means of the method of moving planes in integral forms introduced by Chen-Li-Ou. Thus, the equivalence implies the positive solutions of the PDE system, also have the corresponding properties. This paper extends previous results obtained by other authors to the general case.

2016, 36(7): 3811-3843
doi: 10.3934/dcds.2016.36.3811

*+*[Abstract](1815)*+*[PDF](661.8KB)**Abstract:**

Multi-kink solutions of the defocusing, modified Korteweg-de Vries equation (mKdV) found by Grosse [22,23] are shown to be globally $H^1$-stable, and asymptotically stable. Stability in the one-kink case was previously established by Zhidkov [51] and Merle-Vega [41]. The proof uses transformations linking the mKdV equation with focusing, Gardner-like equations, where stability and asymptotic stability in the energy space are known. We generalize our results by considering the existence, uniqueness and the dynamics of generalized multi-kinks of defocusing, non-integrable gKdV equations, showing the inelastic character of the kink-kink collision in some regimes.

2016, 36(7): 3845-3856
doi: 10.3934/dcds.2016.36.3845

*+*[Abstract](2213)*+*[PDF](359.5KB)**Abstract:**

This paper deals with the global well-posedness of axisymmetric Navier-Stokes equations with swirl. We prove that there exists a global solution of Navier-Stokes equations under some weighted energy for a class of large anisotropic initial data slowly varying in the vertical variable.

2016, 36(7): 3857-3909
doi: 10.3934/dcds.2016.36.3857

*+*[Abstract](1480)*+*[PDF](684.3KB)**Abstract:**

We study time and space equivariant wave maps from $M\times\mathbb{R}\rightarrow S^2,$ where $M$ is diffeomorphic to a two dimensional sphere and admits an action of $SO(2)$ by isometries. We assume that metric on $M$ can be written as $dr^2+f^2(r)d\theta^2$ away from the two fixed points of the action, where the curvature is positive, and prove that stationary (time equivariant) rotationally symmetric (of any rotation number) smooth wave maps exist and are stable in the energy topology. The main new ingredient in the construction, compared with the case where $M$ is isometric to the standard sphere (considered by Shatah and Tahvildar-Zadeh [34]), is the the use of triangle comparison theorems to obtain pointwise bounds on the fundamental solution on a curved background.

2016, 36(7): 3911-3925
doi: 10.3934/dcds.2016.36.3911

*+*[Abstract](1859)*+*[PDF](390.4KB)**Abstract:**

In this paper we establish a closing property and a hyperbolic closing property for thin trapped chain hyperbolic homoclinic classes with one dimensional center in partial hyperbolicity setting. Taking advantage of theses properties, we prove that the growth rate of the number of hyperbolic periodic points is equal to the topological entropy. We also obtain that the hyperbolic periodic measures are dense in the space of invariant measures.

2016, 36(7): 3927-3959
doi: 10.3934/dcds.2016.36.3927

*+*[Abstract](1999)*+*[PDF](697.2KB)**Abstract:**

We construct small-amplitude solitary traveling gravity-capillary water waves with a finite number of point vortices along a vertical line, on finite depth. This is done using a local bifurcation argument. The properties of the resulting waves are also examined: We find that they depend significantly on the position of the point vortices in the water column.

2016, 36(7): 3961-3991
doi: 10.3934/dcds.2016.36.3961

*+*[Abstract](2168)*+*[PDF](568.4KB)**Abstract:**

We introduce a new method for the boundedness problem of semilinear Duffing equations at resonance. In particular, it can be used to study a class of semilinear equations at resonance without the polynomial-like growth condition. As an application, we prove the boundedness of all the solutions for the equation $\ddot{x}+n^2x+g(x)+\psi(x)=p(t)$ under the Lazer-Leach condition on $g$ and $p$, where $n\in \mathbb{N^+}$, $p(t)$ and $\psi(x)$ are periodic and $g(x)$ is bounded.

2016, 36(7): 3993-4014
doi: 10.3934/dcds.2016.36.3993

*+*[Abstract](1885)*+*[PDF](456.6KB)**Abstract:**

We consider open billiards in the plane satisfying the no-eclipse condition. We show that the points in the non-wandering set depend differentiably on deformations to the boundary of the billiard. We use Bowen's equation to estimate the Hausdorff dimension of the non-wandering set of the billiard. Finally we show that the Hausdorff dimension depends differentiably on sufficiently smooth deformations to the boundary of the billiard, and estimate the derivative with respect to such deformations.

2016, 36(7): 4015-4025
doi: 10.3934/dcds.2016.36.4015

*+*[Abstract](1942)*+*[PDF](387.4KB)**Abstract:**

In this paper, we investigate a class of quasi-homogeneous polynomial systems with a given weight degree. Firstly, by some analytical skills, several properties about this kind of systems are derived and an algorithm can be established to obtain all possible explicit systems for a given weight degree. Then, we focus on center problems for such systems and provide some necessary conditions for the existence of centers. Finally, for a specific quasi-homogeneous polynomial system, we characterize its center and prove that the center is not isochronous.

2016, 36(7): 4027-4049
doi: 10.3934/dcds.2016.36.4027

*+*[Abstract](2433)*+*[PDF](488.5KB)**Abstract:**

This paper is concerned with the principal eigenvalues of some nonlocal operators. We first derive a result on the limit of certain sequences of principal eigenvalues associated with some nonlocal eigenvalue problems. Then, such a result is used to study the existence, uniqueness and asymptotic behavior of positive solutions to a nonlocal stationary problem with a parameter. Finally, the long-time behavior of the solutions of the corresponding nonlocal evolution equation and the asymptotic behavior of positive stationary solutions on parameter are discussed.

2016, 36(7): 4051-4062
doi: 10.3934/dcds.2016.36.4051

*+*[Abstract](1958)*+*[PDF](342.6KB)**Abstract:**

In this paper, we first give the explicit variational structure of the nonlinear elastic waves for isotropic, homogeneous, hyperelastic materials in 2-D. Based on this variational structure, we suggest a null condition which is a kind of structural condition on the nonlinearity in order to stop the formation of finite time singularities of local smooth solutions. In the radial symmetric case, inspired by Alinhac's work on 2-D quasilinear wave equations [S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001) 597--618], we show that such null condition can ensure the global existence of smooth solutions with small initial data.

2016, 36(7): 4063-4075
doi: 10.3934/dcds.2016.36.4063

*+*[Abstract](2011)*+*[PDF](359.7KB)**Abstract:**

In this paper we establish a formula of conditional entropy and give two examples of applications of the formula.

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