American Institute of Mathematical Sciences

In this paper, we are concerned with the following high order degenerate elliptic system:

\begin{document}$\left\{ \begin{align} & {{(-A)}^{m}}u={{v}^{p}} \\ & {{(-A)}^{m}}v={{u}^{q}}\quad \text{ in }\mathbb{R}_{+}^{n+1}:=\left\{ (x,y)|x\in {{\mathbb{R}}^{n}},y>0 \right\}, \\ & u\ge 0,v\ge 0 \\ \end{align} \right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$\end{document}

where the operator \begin{document}$ A: = y\partial_{y}^2+a\partial_{y}+\Delta_{x}, \;a\geq 1 $\end{document} and \begin{document}$ n+2a>2m, m\in \mathbb{Z}^+,\;p,\,q\geq 1 $\end{document}. We prove the non-existence of positive smooth solutions for \begin{document}$ 1<p,\, q<\frac{n+2a+2m}{n+2a-2m} $\end{document}, and classify positive solutions for \begin{document}$ p = q = \frac{n+2a+2m}{n+2a-2m} $\end{document}. For \begin{document}$ \frac{1}{p+1}+\frac{1}{q+1}>\frac{n+2a-2m}{n+2a} $\end{document}, we show the non-existence of positive, ellipse-radial, smooth solutions. Moreover we prove the non-existence of positive smooth solutions for the high order degenerate elliptic system of inequalities \begin{document}$ (-A)^{m}u\geq v^p, (-A)^{m}v\geq u^q, u\geq 0, v\geq 0, $\end{document} in \begin{document}$ \mathbb{R}_+^{n+1} $\end{document} for either \begin{document}$ (n+2a-2m)q<\frac{n+2a}{p}+2m $\end{document} or \begin{document}$ (n+2a-2m)p<\frac{n+2a}{q}+2m $\end{document} with \begin{document}$ p,q>1 $\end{document}.

We consider a commutative diagram (CD) of flows with discrete phase group \begin{document}$ T $\end{document} and extensions as follows:

By \begin{document}$ {\text{RP}}_{\!\phi} $\end{document} and \begin{document}$ {\text{RP}}_{\!\psi} $\end{document} we denote the relativized regionally proximal relations in \begin{document}$ X $\end{document} and \begin{document}$ Y $\end{document}, respectively. We mainly prove, among other things, the following:

1. \begin{document}$ $\end{document}If \begin{document}$ X $\end{document} is topologically transitive \begin{document}$ \phi $\end{document}-distal, then \begin{document}$ X $\end{document} is minimal.

2. \begin{document}$ (\pi\times\pi) {\text{RP}}_{\!\phi} = {\text{RP}}_{\!\psi} $\end{document}.

3. If \begin{document}$ Y $\end{document} is locally \begin{document}$ \psi $\end{document}-Bronstein, then \begin{document}$ {\text{RP}}_{\!\psi}\circ {\text{P}}_{\!\psi} $\end{document} is an equivalence relation, and, \begin{document}$ \bar{y}\in {\text{RP}}_{\!\psi} $\end{document} whenever \begin{document}$ \bar{y}\in {\text{RP}}_{\!\psi}\circ {\text{RP}}_{\!\psi} $\end{document} is almost periodic.

4. If \begin{document}$ Y_d $\end{document} is the maximal distal extension of \begin{document}$ Z $\end{document} below \begin{document}$ Y $\end{document} and \begin{document}$ Z^d $\end{document} is the universal minimal distal extension of \begin{document}$ Z $\end{document}, then \begin{document}$ Y\perp_{Y_d}Z^d $\end{document}.

5. (a) \begin{document}$ Y $\end{document} is locally \begin{document}$ \psi $\end{document}-Bronstein iff \begin{document}$ F^d<F^\prime A $\end{document} where \begin{document}$ A,F,F^d $\end{document} are respectively the Ellis groups of \begin{document}$ Y,Z, Z^d $\end{document}. (b) If \begin{document}$ Y $\end{document} is locally \begin{document}$ \psi $\end{document}-Bronstein and \begin{document}$ K $\end{document} a \begin{document}$ \tau $\end{document}-closed group with \begin{document}$ F^\prime A<K<F $\end{document}, then there is a unique \begin{document}$ Y_{\!K} $\end{document} which is \begin{document}$ \psi_{\!K}^\prime $\end{document}-equicontinuous and has the Ellis group \begin{document}$ K $\end{document}.

We also prove the above theorems in the case \begin{document}$ T $\end{document} is a semigroup. Moreover, we show the following in minimal semiflows:

6. \begin{document}$ Y $\end{document} is \begin{document}$ \psi $\end{document}-distal iff \begin{document}$ \psi $\end{document} has a DE-tower iff there is a least group-like extension \begin{document}$ X $\end{document} via \begin{document}$ \phi $\end{document} (i.e., \begin{document}$ \phi^{-1}\phi x = {\text{Aut}}_\phi(T,X)x $\end{document} for all \begin{document}$ x\in X $\end{document}).

7. \begin{document}$ \psi $\end{document} is group-like iff \begin{document}$ \psi $\end{document} has a G-tower that consists of group extensions and inverse limits.

We establish the global existence of weak martingale solutions to the simplified stochastic Ericksen–Leslie system modeling the nematic liquid crystal flow driven by Wiener-type noises on the two-dimensional bounded domains. The construction of solutions is based on the convergence of Ginzburg–Landau approximations. To achieve such a convergence, we first utilize the concentration-cancellation method for the Ericksen stress tensor fields based on a Pohozaev type argument, and then the Skorokhod compactness theorem, which is built upon uniform energy estimates.

By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as \begin{document}$ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $\end{document} with zero boundary data, have unexpected degenerate nature.

We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in \begin{document}$ {{\bf R}}^{n} $\end{document}, and study the asymptotic profile and optimal decay rates of solutions as \begin{document}$ t \to \infty $\end{document} in \begin{document}$ L^{2} $\end{document}-sense. The operator \begin{document}$ L $\end{document} considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charão-Ikehata [7]. We will discuss the asymptotic property of the solution as time goes to infinity to our Cauchy problem, and in particular, we classify the property of the solutions into three parts from the viewpoint of regularity of the initial data, that is, diffusion-like, wave-like, and both of them.

In this paper we examine the well-posedness and ill-posedeness of the Cauchy problems associated with a family of equations of ZK-KP-type

\begin{document}$ \begin{cases} u_{t} = u_{xxx}-\mathscr{H}D_{x}^{\alpha}u_{yy}+uu_{x}, \cr u(0) = \psi \in Z \end{cases} $\end{document}

in anisotropic Sobolev spaces, where \begin{document}$ 1\le \alpha \le 1 $\end{document}, \begin{document}$ \mathscr{H} $\end{document} is the Hilbert transform and \begin{document}$ D_{x}^{\alpha} $\end{document} is the fractional derivative, both with respect to \begin{document}$ x $\end{document}.

We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces. In particular, the abstract weak orbit spaces of flows on topological spaces are refinements of Morse graphs of flows on compact metric spaces, Reeb graphs of Hamiltonian flows with finitely many singular points on surfaces, and the CW decompositions which consist of the unstable manifolds of singular points for Morse flows on closed manifolds. Though the CW decomposition of a Morse flow is finite, the intersection of the unstable manifold and the stable manifold of closed orbits need not consist of finitely many connected components. Therefore we study the finiteness. Moreover, we consider when the time-one map reconstructs the topology of the original flow. We show that the orbit space of a Hamiltonian flow with finitely many singular points on a compact surface is homeomorphic to the abstract weak orbit space of the time-one map by taking an arbitrarily small reparametrization and that the abstract weak orbit spaces of a Morse flow on a compact manifold and the time-one map are homeomorphic. In addition, we state examples whose Morse graphs are singletons but whose abstract weak orbit spaces are not singletons.

In this paper, we consider regularity criteria of a class of 3D axially symmetric MHD-Boussinesq systems without magnetic resistivity or thermal diffusivity. Under some Prodi-Serrin type critical assumptions on the horizontal angular component of the velocity, we will prove that strong solutions of the axially symmetric MHD-Boussinesq system can be smoothly extended beyond the possible blow-up time \begin{document}$ T_\ast $\end{document} if the magnetic field contains only the horizontal swirl component. No a priori assumption on the magnetic field or the temperature fluctuation is imposed.

We present a mechanism for the emergence of strange attractors in a one-parameter family of differential equations defined on a 3-dimensional sphere. When the parameter is zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two saddles-foci with different Morse indices. After slightly increasing the parameter, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We will show that, for a set of parameters close enough to zero with positive Lebesgue measure, the dynamics exhibits strange attractors winding around the "ghost'' of a torus and supporting Sinai-Ruelle-Bowen (SRB) measures. We also prove the existence of a sequence of parameter values for which the family exhibits a superstable sink and describe the transition from a Bykov network to a strange attractor.

In this paper, we prove the existence of traveling wave solutions for an incompressible Darcy's free boundary problem recently introduced in [6] to describe cell motility. This free boundary problem involves a nonlinear destabilizing term in the boundary condition which describes the active character of the cell cytoskeleton. By using two different methods, a constructive method via a graph analysis and a local bifurcation method, we prove that traveling wave solutions exist when the destabilizing term is strong enough.

We study the flocking model for continuous time introduced by Cucker and Smale adding a positive time delay \begin{document}$ \tau $\end{document}. The goal of this article is to prove that the same unconditional flocking result for the non-delayed case is valid in the delayed case. A novelty is that we do not need to impose any restriction on the size of \begin{document}$ \tau $\end{document}. Furthermore, when the unconditional flocking occurs, velocities converge exponentially fast to a common one.

We add a lemma implicitly used in the proof of the forward Ergodic Closing Lemma in the paper "A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies" [Discrete Contin. Dyn. Syst., 40 (2020), 2285-2313].

We demonstrate that there is a large class of compact metric spaces for which the shadowing property can be characterized as a structural property of the space of dynamical systems. We also demonstrate that, for this class of spaces, in order to determine whether a system has shadowing, it is sufficient to check that continuously generated pseudo-orbits can be shadowed.

We introduce a regularization-free approach for the wellposedness of the classic Cahn-Hilliard equation with logarithmic potentials.

In this paper, we study the topological spectrum of weighted Birk–hoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave over its domain. In case of typical weights with respect to some ergodic quasi-Bernoulli measure, we determine the spectrum. Moreover, in case of full shift and under the assumption that the potentials depend only on the first coordinate, we show that our result is applicable for regular weights, like Möbius sequence.

One of the most impressive findings in chemotaxis is the aggregation that randomly distributed bacteria, when starved, release a diffusive chemical to attract and group with others to form one or several stable aggregates in a long time. This paper considers pattern formation within the minimal Keller–Segel chemotaxis model with a focus on the stability and dynamics of its multi-spike steady states. We first show that any steady-state must be a periodic replication of the spatially monotone one and they present multi-spikes when the chemotaxis rate is large; moreover, we prove that all the multi-spikes are unstable through their refined asymptotic profiles, and then find a fully-fledged hierarchy of free entropy energy of these aggregates. Our results also complement the literature by finding that when the chemotaxis is strong, the single boundary spike has the least energy hence is the most stable, the steady-state with more spikes has larger free energy, while the constant has the largest free energy and is always unstable. These results provide new insights into the model's intricate global dynamics, and they are illustrated and complemented by numerical studies which also demonstrate the metastability and phase transition behavior in chemotactic movement.

For \begin{document}$ C^2 $\end{document} cos-type potentials, large coupling constants, and fixed \begin{document}$ Diophantine $\end{document} frequency, we show that the density of the spectral points associated with the Schrödinger operator is larger than 0. In other words, for every fixed spectral point \begin{document}$ E $\end{document}, \begin{document}$ \liminf\limits_{\epsilon\to 0}\frac{|(E-\epsilon,E+\epsilon)\bigcap\Sigma_{\alpha,\lambda\upsilon}|}{2\epsilon} = \beta $\end{document}, where \begin{document}$ \beta\in [\frac{1}{2},1] $\end{document}. Our approach is a further improvement on the papers [15] and [17].