# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

January 2018 , Volume 38 , Issue 1

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2018, 38(1): 1-41 doi: 10.3934/dcds.2018001 +[Abstract](3697) +[HTML](104) +[PDF](1304.6KB)
Abstract:

We present a new strategy for proving the Ambrose conjecture, a global version of the Cartan local lemma. We introduce the concepts of linking curves, unequivocal sets and sutured manifolds, and show that any sutured manifold satisfies the Ambrose conjecture. We then prove that the set of sutured Riemannian manifolds contains a residual set of the metrics on a given smooth manifold of dimension $3$.

2018, 38(1): 43-62 doi: 10.3934/dcds.2018002 +[Abstract](6103) +[HTML](190) +[PDF](809.1KB)
Abstract:

In this paper, we investigate diffusive predator-prey systems with nonlocal intraspecific competition of prey for resources. We prove the existence and uniqueness of positive steady states when the conversion rate is large. To show the existence of complex spatiotemporal patterns, we consider the Hopf bifurcation for a spatially homogeneous kernel function, by using the conversion rate as the bifurcation parameter. Our results suggest that Hopf bifurcation is more likely to occur with nonlocal competition of prey. Moreover, we find that the steady state can lose the stability when conversion rate passes through some Hopf bifurcation value, and the bifurcating periodic solutions near such bifurcation value can be spatially nonhomogeneous. This phenomenon is different from that for the model without nonlocal competition of prey, where the bifurcating periodic solutions are spatially homogeneous near such bifurcation value.

2018, 38(1): 63-74 doi: 10.3934/dcds.2018003 +[Abstract](3405) +[HTML](84) +[PDF](387.0KB)
Abstract:

Let \begin{document} $B_{\alpha ,a}$ \end{document} be the Blaschke product of the following form:

If \begin{document} $B_{\alpha ,a}|_{S^1}$ \end{document} is analytically linearizable, then there is a Herman ring admitting the unit circle as an invariant curve in the dynamical plane of \begin{document} $B_{\alpha ,a}$ \end{document}. Given an irrational number \begin{document} $θ$ \end{document}, the parameters \begin{document} $(\alpha ,a)$ \end{document} such that \begin{document} $B_{\alpha ,a}|_{S^1}$ \end{document} has rotation number \begin{document} $θ$ \end{document} form a curve \begin{document} $T_d(θ)$ \end{document} in the parameter plane. Using quasiconformal surgery, we prove that if \begin{document} $θ$ \end{document} is of Brjuno type, the curve can be parameterized real analytically by the modulus of the Herman ring, from \begin{document} $a=M(θ)$ \end{document} up to \begin{document} $∞$ \end{document} with \begin{document} $M(θ)≥q 2d+1$ \end{document}, for which the Herman ring vanishes.Moreover, we can show that for a certain set of irrational numbers \begin{document} $θ ∈ \mathcal {B}\setminus\mathcal {H}$ \end{document}, the number \begin{document} $M(θ)$ \end{document} is strictly greater than \begin{document} $2d+1$ \end{document} and the boundary of the Herman rings consist of two quasicircles not containing any critical point.

2018, 38(1): 75-90 doi: 10.3934/dcds.2018004 +[Abstract](4181) +[HTML](104) +[PDF](412.7KB)
Abstract:

In this paper, we study regularity of solutions of elliptic systems in divergence form with directional homogenization. Here directional homogenization means that the coefficients of equations are rapidly oscillating only in some directions. We will investigate the different regularity of solutions on directions with homogenization and without homogenization. Actually, we obtain uniform interior \begin{document}$W^{1, p}$\end{document} estimates in all directions and uniform interior \begin{document}$C^{1, γ}$\end{document} estimates in the directions without homogenization.

2018, 38(1): 91-129 doi: 10.3934/dcds.2018005 +[Abstract](3279) +[HTML](94) +[PDF](674.0KB)
Abstract:

Let $n≥ 3$ and $m=\frac{n-2}{n+2}$. We construct $5$-parameters, $4$-parameters, and $3$-parameters ancient solutions of the equation $v_t=(v^m)_{xx}+v-v^m$, $v>0$, in $\mathbb{R}× (-∞, T)$ for some $T∈\mathbb{R}$. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient solutions of this equation including exact decay rate of ancient solutions as $|x|\to∞$. We also prove that both the $3$-parameters ancient solution and the $4$-parameters ancient solution are singular limit solution of the $5$-parameters ancient solutions. We also prove the uniqueness of the $4$-parameters ancient solutions. As a consequence we prove that the $4$-parameters ancient solutions that we construct coincide with the $4$-parameters ancient solutions constructed by P. Daskalopoulos, M. del Pino, J. King, and N. Sesum in [8].

2018, 38(1): 131-154 doi: 10.3934/dcds.2018006 +[Abstract](3854) +[HTML](90) +[PDF](479.0KB)
Abstract:

In this paper we are concerned with the existence of invariant curves of planar mappings which are quasi-periodic in the spatial variable, satisfy the intersection property, $\mathcal{C}^{p}$ smooth with $p>2n+1$, $n$ is the number of frequencies.

2018, 38(1): 155-168 doi: 10.3934/dcds.2018007 +[Abstract](3535) +[HTML](85) +[PDF](381.0KB)
Abstract:

We apply lattice points counting results to solve a shrinking target problem in the setting of discrete time geodesic flows on hyperbolic manifolds of finite volume.

2018, 38(1): 169-186 doi: 10.3934/dcds.2018008 +[Abstract](4022) +[HTML](84) +[PDF](507.3KB)
Abstract:

For a local field K of formal Laurent series and its ring Z of polynomials, we prove a pointwise equidistribution with an error rate of each H-orbit in SL(d, K)/SL(d, Z) for a certain proper subgroup H of a horospherical group, extending a work of Kleinbock-Shi-Weiss.

We obtain an asymptotic formula for the number of integral solutions to the Diophantine inequalities with weights, generalizing a result of Dodson-Kristensen-Levesley. This result enables us to show pointwise equidistribution for unbounded functions of class Cα.

2018, 38(1): 187-208 doi: 10.3934/dcds.2018009 +[Abstract](4609) +[HTML](102) +[PDF](474.9KB)
Abstract:

In this paper, we study the limiting behavior of dynamics for stochastic reaction-diffusion equations driven by an additive noise and a deterministic non-autonomous forcing on an (n+1)-dimensional thin region when it collapses into an n-dimensional region. We first established the existence of attractors and their properties for these equations on (n+1)-dimensional thin domains. We then show that these attractors converge to the random attractor of the limit equation under the usual semi-distance as the thinness goes to zero.

2018, 38(1): 209-230 doi: 10.3934/dcds.2018010 +[Abstract](3903) +[HTML](60) +[PDF](472.8KB)
Abstract:

In this work, we prove single-point blow-up for any positive, radially decreasing, classical and blowing-up solution of a system of \begin{document}$m≥q3$\end{document} heat equations in a ball of \begin{document}$\mathbb{R}^n$\end{document}, which are coupled cyclically by superlinear monomial reaction terms. We also obtain lower pointwise estimates for the blow-up profiles.

2018, 38(1): 231-245 doi: 10.3934/dcds.2018011 +[Abstract](3299) +[HTML](71) +[PDF](430.1KB)
Abstract:

In the previous investigations of the authors the renormalization group method to p-adic models on Cayley trees has been developed. This method is closely related to the investigation of p-adic dynamical systems associated with a given model. In this paper, we study chaotic behavior of the Potts-Bethe mapping. We point out that a similar kind of result is not known in the case of real numbers (with rigorous proofs).

2018, 38(1): 247-261 doi: 10.3934/dcds.2018012 +[Abstract](3301) +[HTML](62) +[PDF](407.0KB)
Abstract:

We show that given any \begin{document}$C^{\infty}$\end{document} Riemannian structure \begin{document}$(T^{2},g)$\end{document} in the two torus, \begin{document}$\epsilon >0$\end{document} and \begin{document}$\beta \in (0,\frac{1}{3})$\end{document}, there exists a \begin{document}$C^{\infty}$\end{document} Riemannian metric \begin{document}$\bar{g}$\end{document} with no continuous Lagrangian invariant graphs that is \begin{document}$\epsilon$\end{document}-\begin{document}$C^{1,\beta}$\end{document} close to \begin{document}$g$\end{document}. The main idea of the proof is inspired in the work of V. Bangert who introduced caps from smoothed cone type \begin{document}$C^{1}$\end{document} small perturbations of metrics with non-positive curvature to get conjugate points. Our new contribution to the subject is to show that positive curvature cone type small perturbations are less singular" than non-positive curvature cone type perturbations. Positive curvature geometry allows us to get better estimates for the variation of the \begin{document}$C^{1}$\end{document} norm of the singular cone in a neighborhood of its vertex.

2018, 38(1): 263-292 doi: 10.3934/dcds.2018013 +[Abstract](3870) +[HTML](64) +[PDF](458.6KB)
Abstract:

We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these general conditions to derive results for spectral properties of time-changes of unipotent flows on homogeneous spaces of semisimple groups regarding absolutely continuous spectrum as well as maximal spectral type; the time-changes of the horocycle flow are special cases of this general category of flows. In addition we use the general conditions to derive spectral results for twisted horocycle flows and to rederive certain spectral results for skew products over translations and Furstenberg transformations.

2018, 38(1): 293-309 doi: 10.3934/dcds.2018014 +[Abstract](4185) +[HTML](91) +[PDF](392.2KB)
Abstract:

The method of upper and lower solutions is a main tool to prove the existence of periodic solutions to periodic differential equations. It is known that, in general, the method does not extend to the almost periodic case. The aim of the present paper is to show that, however, something interesting survives: if the classical assumptions of the method are satisfied, then the expected existence result holds generically in the limit periodic framework.

2018, 38(1): 311-328 doi: 10.3934/dcds.2018015 +[Abstract](3671) +[HTML](74) +[PDF](452.36KB)
Abstract:

In this paper, we consider the Cauchy problem of the rotating shallow water equations, which has height-dependent viscosities, arbitrarily large initial data and far field vacuum. Firstly, we establish the existence of the unique local regular solution, whose life span is uniformly positive as the viscosity coefficients vanish. Secondly, we prove the well-posedness of the regular solution for the inviscid flow. Finally, we show the convergence rate of the regular solution from the viscous flow to the inviscid flow in \begin{document}$L^{\infty}([0, T]; H^{s'})$\end{document} for any \begin{document}$s'\in [2, 3)$\end{document} with a rate of \begin{document}$\epsilon^{1-\frac{s'}{3}}$\end{document}.

2018, 38(1): 329-341 doi: 10.3934/dcds.2018016 +[Abstract](3823) +[HTML](81) +[PDF](418.8KB)
Abstract:

In this paper, we study symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system. Lie symmetry analysis and similarity reductions are performed, some invariant solutions are also discussed. Then prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. Furthermore, we show that the system exhibits unique continuation if the initial momentum \begin{document}$m_0$\end{document} and \begin{document}$n_0$\end{document} are positive.

2018, 38(1): 343-361 doi: 10.3934/dcds.2018017 +[Abstract](3836) +[HTML](65) +[PDF](425.1KB)
Abstract:

We are interested in the asymptotic behaviour of the first return time of the orbits of a dynamical system into a small neighbourhood of their starting points. We study this quantity in the context of dynamical systems preserving an infinite measure. More precisely, we consider the case of \begin{document}$\mathbb{Z}$\end{document}-extensions of subshifts of finite type. We also consider a toy probabilistic model to enlight the strategy of our proofs.

2018, 38(1): 363-396 doi: 10.3934/dcds.2018018 +[Abstract](3958) +[HTML](76) +[PDF](656.6KB)
Abstract:

We study time asymptotic decay of solutions for a general system of hyperbolic-parabolic balance laws in multi space dimensions. The system has physical viscosity matrices and a lower order term for relaxation, damping or chemical reaction. The viscosity matrices and the Jacobian matrix of the lower order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper we obtain optimal $L^p$ decay rates for $p≥2$. Our result is general and applies to physical models such as gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively.

2018, 38(1): 397-429 doi: 10.3934/dcds.2018019 +[Abstract](3661) +[HTML](63) +[PDF](546.68KB)
Abstract:

The focus of the current paper is the higher order nonlinear dispersive equation which models unidirectional propagation of small amplitude long waves in dispersive media. The specific interest is in the initial-boundary value problem where spatial variable lies in \begin{document}$\mathbb R^+,$\end{document} namely, quarter plane problem. With proper requirement on initial and boundary condition, we show local and global well posedness.

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