# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems

April 2017 , Volume 37 , Issue 4

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2017, 37(4): 1789-1818 doi: 10.3934/dcds.2017075 +[Abstract](2812) +[HTML](65) +[PDF](553.5KB)
Abstract:

We consider the following mean field type equations on domains of \begin{document}$\mathbb R^2$\end{document} under Dirichlet boundary conditions:

where \begin{document} $K$ \end{document} is a smooth positive function and \begin{document} $\varrho$ \end{document} is a positive real parameter.

A "critical point theory at Infinity" approach of A. Bahri to the above problem is developed for the resonant case, i.e. when the parameter \begin{document} $\varrho$ \end{document} is a multiple of \begin{document} $8 π$ \end{document}. Namely, we identify the so-called "critical points at infinity" of the associated variational problem and compute their Morse indices. We then prove some Bahri-Coron type results which can be seen as a generalization of a degree formula in the non-resonant case due to C.C.Chen and C.S.[18].

2017, 37(4): 1819-1839 doi: 10.3934/dcds.2017076 +[Abstract](2444) +[HTML](55) +[PDF](230.2KB)
Abstract:

Let \begin{document} $(S,d)$ \end{document} be a compact metric space and let \begin{document} $m$ \end{document} be a Borel probability measure on \begin{document} $(S,d)$ \end{document}. We shall prove that, if \begin{document} $(S,d,m)$ \end{document} is a \begin{document} $RCD(K,\infty)$ \end{document} space, then the stochastic value function satisfies the viscous Hamilton-Jacobi equation, exactly as in Fleming's theorem on \begin{document} ${\bf{R}}^d$ \end{document}.

2017, 37(4): 1841-1856 doi: 10.3934/dcds.2017077 +[Abstract](2496) +[HTML](66) +[PDF](424.5KB)
Abstract:

We consider two-dimensional versions of the Keller-Segel model for the chemotaxis with either classical (Brownian) or fractional (anomalous) diffusion. Criteria for blowup of solutions in terms of suitable Morrey spaces norms are derived. Moreover, the impact of the consumption term on the global-in-time existence of solutions is analyzed for the classical Keller-Segel system.

2017, 37(4): 1857-1865 doi: 10.3934/dcds.2017078 +[Abstract](2668) +[HTML](58) +[PDF](402.7KB)
Abstract:

We prove bifurcation at infinity for a semilinear wave equation depending on a parameter \begin{document}$λ$\end{document} and subject to Dirichlet-periodic boundary conditions. We assume the nonlinear term to be asymptotically linear and not necessarily monotone. We prove the existence of L solutions tending to \begin{document}$+∞$\end{document} when the bifurcation parameter approaches eigenvalues of finite multiplicity of the wave operator. Further details are presented in cases of simple eigenvalues and odd multiplicity eigenvalues.

2017, 37(4): 1867-1901 doi: 10.3934/dcds.2017079 +[Abstract](2445) +[HTML](56) +[PDF](548.6KB)
Abstract:

The existence of 2-dimensional KAM tori is proved for the perturbed generalized nonlinear vibrating string equation with singularities \begin{document} $u_{tt}=((1-x^2)u_x)_x-mu-u^3$ \end{document} subject to certain boundary conditions by means of infinite-dimensional KAM theory with the help of partial Birkhoff normal form, the characterization of the singular function space and the estimate of the integrals related to Legendre basis.

2017, 37(4): 1903-1922 doi: 10.3934/dcds.2017080 +[Abstract](2565) +[HTML](52) +[PDF](1237.8KB)
Abstract:

We present a proof of the existence of a periodic orbit for the Newtonian six-body problem with equal masses. This orbit has three double collisions each period and no multiple collisions. Our proof is based on the minimization of the lagrangian action functional on a well chosen class of symmetric loops.

2017, 37(4): 1923-1939 doi: 10.3934/dcds.2017081 +[Abstract](3178) +[HTML](91) +[PDF](472.4KB)
Abstract:

In this paper, we study the three-dimensional axisymmetric Navier-Stokes system with nonzero swirl. By establishing a new key inequality for the pair \begin{document}$(\frac{ω^{r}}{r},\frac{ω^{θ}}{r})$\end{document}, we get several Prodi-Serrin type regularity criteria based on the angular velocity, \begin{document}$u^θ$\end{document}. Moreover, we obtain the global well-posedness result if the initial angular velocity \begin{document}$u_{0}^{θ}$\end{document} is appropriate small in the critical space \begin{document}$L^{3}(\mathbb{R}^{3})$\end{document}. Furthermore, we also get several Prodi-Serrin type regularity criteria based on one component of the solutions, say \begin{document}$ω^3$\end{document} or \begin{document}$u^3$\end{document}.

2017, 37(4): 1941-1957 doi: 10.3934/dcds.2017082 +[Abstract](2959) +[HTML](71) +[PDF](383.8KB)
Abstract:

We relate the local specification and periodic shadowing properties. We also clarify the relation between local weak specification and local specification if the system is measure expansive. The notion of strong measure expansiveness is introduced, and an example of a non-expansive systems with the strong measure expansive property is given. Moreover, we find a family of examples with the $N$-expansive property, which are not strong measure expansive. We finally show a spectral decomposition theorem for strong measure expansive dynamical systems with shadowing.

2017, 37(4): 1959-1977 doi: 10.3934/dcds.2017083 +[Abstract](2737) +[HTML](60) +[PDF](467.5KB)
Abstract:

Existence of almost automorphic solutions for abstract delayed differential equations is established. Using ergodicity, exponential dichotomy and Bi-almost automorphicity on the homogeneous part, sufficient conditions for the existence and uniqueness of almost automorphic solutions are given.

2017, 37(4): 1979-2007 doi: 10.3934/dcds.2017084 +[Abstract](2969) +[HTML](58) +[PDF](545.8KB)
Abstract:

We prove that for the Cauchy problem of focusing \begin{document}$L^2$\end{document}-critical Hartree equations with spherically symmetric \begin{document}$H^1$\end{document} data in dimensions \begin{document}$3$\end{document} and \begin{document}$4$\end{document}, the global non-scattering solution with ground state mass must be a solitary wave up to symmetries of the equation. The approach is a linearization analysis around the ground state combined with an in-out spherical wave decomposition technique.

2017, 37(4): 2009-2021 doi: 10.3934/dcds.2017085 +[Abstract](2908) +[HTML](61) +[PDF](426.6KB)
Abstract:

We find a simple sufficient criterion on a pair of nonnegative weight functions \begin{document}$V(x)$\end{document} and \begin{document}$W(x)$\end{document} on a Carnot group \begin{document}$\mathbb{G},$\end{document} so that the general weighted \begin{document}$L^{p}$\end{document} Hardy type inequality

is valid for any \begin{document}$φ ∈ C_{0}^{∞ }(\mathbb{G})$\end{document} and \begin{document}$p>1.$\end{document} It is worth noting here that our unifying method may be readily used both to recover most of the previously known weighted Hardy and Heisenberg-Pauli-Weyl type inequalities as well as to construct other new inequalities with an explicit best constant on \begin{document}$\mathbb{G}.$\end{document} We also present some new results on two-weight \begin{document}$L^{p}$\end{document} Hardy type inequalities with remainder terms on a bounded domain \begin{document}$Ω$\end{document} in \begin{document}$\mathbb{G}$\end{document} via a differential inequality.

2017, 37(4): 2023-2043 doi: 10.3934/dcds.2017086 +[Abstract](3350) +[HTML](58) +[PDF](3599.6KB)
Abstract:

In this work we are interested in the mean-field formulation of kinetic models under control actions where the control is formulated through a model predictive control strategy (MPC) with varying horizon. The relation between the (usually hard to compute) optimal control and the MPC approach is investigated theoretically in the mean-field limit. We establish a computable and provable bound on the difference in the cost functional for MPC controlled and optimal controlled system dynamics in the mean-field limit. The result of the present work extends previous findings for systems of ordinary differential equations. Numerical results in the mean-field setting are given.

2017, 37(4): 2045-2063 doi: 10.3934/dcds.2017087 +[Abstract](2744) +[HTML](57) +[PDF](457.4KB)
Abstract:

We are concerned with the global well-posedness of the diffusion approximation model in radiation hydrodynamics, which describe the compressible fluid dynamics taking into account the radiation effect under the non-local thermal equilibrium case. The model consist of the compressible Navier-Stokes equations coupled with the radiative transport equation with non-local terms. Global well-posedness of the Cauchy problem is established in perturbation framework, and rates of convergence of solutions toward equilibrium, which are algebraic in the whole space and exponential on torus, are also obtained under some additional conditions on initial data. The existence of global solution is proved based on the classical energy estimates, which are considerably complicated and some new ideas and techniques are thus required. Moreover, it is shown that neither shock waves nor vacuum and concentration in the solution are developed in a finite time although there is a complex interaction between photons and matter.

2017, 37(4): 2065-2075 doi: 10.3934/dcds.2017088 +[Abstract](2668) +[HTML](62) +[PDF](428.0KB)
Abstract:

Small-amplitude waves in the Fermi-Pasta-Ulam (FPU) lattice with weakly anharmonic interaction potentialsare described by the generalized Korteweg-de Vries (KdV) equation. Justification of the small-amplitudeapproximation is usually performed on the time scale, for which dynamics of the KdV equation is defined.We show how to extend justification analysis on longer time intervals provided dynamics of the generalized KdVequation is globally well-posed in Sobolev spaces and either the Sobolev norms are globally boundedor they grow at most polynomially. The time intervals are extended respectively by the logarithmic or double logarithmic factorsin terms of the small amplitude parameter. Controlling the approximation error on longer time intervalsallows us to deduce nonlinear metastability of small FPU solitary waves from orbital stability of the KdV solitary waves.

2017, 37(4): 2077-2102 doi: 10.3934/dcds.2017089 +[Abstract](3072) +[HTML](64) +[PDF](567.6KB)
Abstract:

We consider non-gauge-invariant cubic nonlinear Schrödinger equations in one space dimension.We show that initial data of size \begin{document}$\varepsilon$\end{document} in a weighted Sobolev space lead to solutions with sharp \begin{document}$L_x^∞$\end{document} decay up to time \begin{document}$\exp(C\varepsilon^{-2})$\end{document}. We also exhibit norm growth beyond this time for a specific choice of nonlinearity.

2017, 37(4): 2103-2113 doi: 10.3934/dcds.2017090 +[Abstract](2515) +[HTML](64) +[PDF](367.1KB)
Abstract:

This paper is devoted to the study of the two-dimensional andthree-dimensional ideal incompressible magneto-hydrodynamic (MHD)equations in which the Faraday law is inviscid. We consider thelocal existence and uniqueness of classical solutions for the MHDsystem in Hölder space when the general initial data belongs to\begin{document}$C^{1,α}(\mathbb{R}^n)$\end{document} for \begin{document}$n=2$\end{document} and \begin{document}$n=3$\end{document}.

2017, 37(4): 2115-2140 doi: 10.3934/dcds.2017091 +[Abstract](3342) +[HTML](1783) +[PDF](813.0KB)
Abstract:

In this work we study degenerate with respect to parameters fold-Hopfbifurcations in three-dimensional differential systems. Such degeneraciesarise when the transformations between parameters leading to a normal formare not regular at some points in the parametric space. We obtain newgeneric results for the dynamics of the systems in such a degenerateframework. The bifurcation diagrams we obtained show that in a degeneratecontext the dynamics may be completely different than in a non-degenerateframework.

2017, 37(4): 2141-2160 doi: 10.3934/dcds.2017092 +[Abstract](2797) +[HTML](55) +[PDF](489.6KB)
Abstract:

In this paper we consider the persistence of elliptic lower dimensional invariant tori with one normal frequency in reversible systems, andprove that if the frequency mapping \begin{document}$ω(y) ∈ \mathbb{R}^n$\end{document} and normal frequency mapping \begin{document}$λ(y) ∈ \mathbb{R}$\end{document} satisfy that

where \begin{document}$ω_0 =ω(y_0)$\end{document} and \begin{document}$λ_0 = λ(y_0)$\end{document} satisfy Melnikov's non-resonance conditions for some \begin{document}$y_0∈\mathcal{O}$\end{document}, then the direction of this frequency for the invariant torus persists under small perturbations. Our result is a generalization of X. Wang et al[Persistence of lower dimensional elliptic invariant tori for a class of nearly integrablereversible systems, Discrete and Continuous Dynamical Systems series B, 14 (2010), 1237-1249].

2017, 37(4): 2161-2180 doi: 10.3934/dcds.2017093 +[Abstract](2564) +[HTML](52) +[PDF](508.8KB)
Abstract:

In this paper, we study the existence-uniqueness and exponential estimate of the pathwise mild solution of retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. Firstly, the existence-uniqueness of the maximal local pathwise mild solution are given by the generalized local Lipschitz conditions, which extend a classical Pazy theorem on PDEs. We assume neither that the noise is given in additive form or that it is a very simple multiplicative noise, nor that the drift coefficient is global Lipschitz continuous. Secondly, the existence-uniqueness of the global pathwise mild solution are given by establishing an integral comparison principle, which extends the classical Wintner theorem on ODEs. Thirdly, an exponential estimate for the pathwise mild solution is obtained by constructing a delay integral inequality. Finally, the results obtained are applied to a retarded stochastic infinite system and a stochastic partial functional differential equation. Combining some known results, we can obtain a random attractor, whose condition overcomes the disadvantage in existing results that the exponential converging rate is restricted by the maximal admissible value for the time delay.

2017, 37(4): 2181-2205 doi: 10.3934/dcds.2017094 +[Abstract](2878) +[HTML](66) +[PDF](540.7KB)
Abstract:

The paper investigates the existence of global attractor for a strongly damped wave equation with fully supercritical nonlinearities: \begin{document}$u_{tt}-Δ u- Δu_t+h(u_t)+g(u)=f(x)$\end{document}. In the case when the nonlinearities \begin{document}$h(u_t)$\end{document} and \begin{document}$g(u)$\end{document} are of fully supercritical growth, which leads to that the weak solutions of the equation lose their uniqueness, by introducing the notion of limit solutions and using the theory on the attractor of the generalized semiflow, we establish the existence of global attractor for the subclass of limit solutions of the equation in natural energy space in the sense of strong topology.

2017, 37(4): 2207-2226 doi: 10.3934/dcds.2017095 +[Abstract](2414) +[HTML](58) +[PDF](493.2KB)
Abstract:

We investigate the followingDirichlet problem with variable exponents:

We present here, in the system setting, a new set of growth conditions under which we manage to use a novel method to verify the Cerami compactness condition. By localization argument, decomposition technique and variational methods, we are able to show the existence of multiple solutions with constant sign for the problem without the well-knownAmbrosetti-Rabinowitz type growth condition. More precisely, we manage to show that the problem admitsfour, six and infinitely many solutions respectively.

2017, 37(4): 2227-2242 doi: 10.3934/dcds.2017096 +[Abstract](2361) +[HTML](56) +[PDF](528.6KB)
Abstract:

In this paper we proved that under the Lazutkin's coordinate, the billiard map can be interpolated by a time-1 flow of a Hamiltonian \begin{document}$H(x,p,t)$\end{document} which can be formally expressed by

where \begin{document}$V(·,·,·)$\end{document} is \begin{document}$C^{r-5}$\end{document} smooth if the convex billiard boundary is \begin{document}$C^r$\end{document} smooth. Benefit from this suspension we can construct transitive trajectories between two adjacent caustics under a variational framework.

2017, 37(4): 2243-2257 doi: 10.3934/dcds.2017097 +[Abstract](2383) +[HTML](59) +[PDF](394.0KB)
Abstract:

The rotation-two-componentCamassa-Holm system with the effect of the Coriolis force in therotating fluid is a model in the equatorial water waves. In thispaper we consider its periodic Cauchy problem. The precise blow-upscenarios of strong solutions and several conditions on the initialdata that produce blow-up of the induced solutions are described indetail. Finally, a sufficient condition for global solutions isestablished.

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