American Institute of Mathematical Sciences

We investigate the large time behavior of \begin{document}$ N $\end{document} particles restricted to a smooth closed curve in \begin{document}$ \mathbb{R}^d $\end{document} and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz \begin{document}$ s $\end{document}-energy with \begin{document}$ s>1. $\end{document} We show that regardless of their initial positions, for all \begin{document}$ N $\end{document} and time \begin{document}$ t $\end{document} large, their normalized Riesz \begin{document}$ s $\end{document}-energy will be close to the \begin{document}$ N $\end{document}-point minimal possible energy. Furthermore, the distribution of such particles will be close to uniform with respect to arclength measure along the curve.

In this paper, we give a positive answer to the problem that whether one can identify the shape of a right triangle billiard table by a single bounce sequence. Moreover, a convenient method to calculate the shape of polygons is given in this paper, too.

In this paper, we study the following nonlinear magnetic Kirchhoff equation with critical growth

\begin{document}$ \begin{align*} \left\{ \begin{aligned} &-\Big(a\epsilon^{2}+b\epsilon\, [u]_{A/\epsilon}^{2}\Big)\Delta_{A/\epsilon} u+V(x)u = f(|u|^{2})u+\vert u\vert^{4}u \quad \hbox{in }\mathbb{R}^3, \\ &u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}), \end{aligned} \right. \end{align*} $\end{document}

where \begin{document}$ \epsilon>0 $\end{document} is a parameter, \begin{document}$ a, b>0 $\end{document} are constants, \begin{document}$ V:\mathbb{R}^{3}\rightarrow \mathbb{R} $\end{document} and \begin{document}$ A: \mathbb{R}^{3}\rightarrow \mathbb{R}^{3} $\end{document} are continuous potentials, and \begin{document}$ f: \mathbb{R}\rightarrow \mathbb{R} $\end{document} is a nonlinear term with subcritical growth. Under a local assumption on the potential \begin{document}$ V $\end{document}, combining variational methods, penalization techniques and the Ljusternik-Schnirelmann theory, we establish multiplicity and concentration properties of solutions to the above problem for \begin{document}$ \varepsilon $\end{document} small. A feature of this paper is that the function \begin{document}$ f $\end{document} is assumed to be only continuous, which allows to consider larger classes of nonlinearities in the reaction.

We investigate systematically several topological transitivity and mixing concepts for group actions via weak disjointness, return time sets and topological complexity functions.

In this paper, we study the two phase flow problem in the ideal incompressible magnetohydrodynamics. We propose a Syrovatskij type stability condition, and prove the local well-posedness of the two phase flow problem with initial data satisfies such condition. This result shows that the magnetic field has a stabilizing effect on Kelvin-Helmholtz instability even the fluids on each side of the free interface have different densities.

In this paper, we explore the period tripling and period quintupling renormalizations below \begin{document}$ C^2 $\end{document} class of unimodal maps. We show that for a given proper scaling data there exists a renormalization fixed point on the space of piece-wise affine maps which are infinitely renormalizable. Furthermore, we show that this renormalization fixed point is extended to a \begin{document}$ C^{1+Lip} $\end{document} unimodal map, considering the period tripling and period quintupling combinatorics. Moreover, we show that there exists a continuum of fixed points of renormalizations by considering a small variation on the scaling data. Finally, this leads to the fact that the tripling and quintupling renormalizations acting on the space of \begin{document}$ C^{1+Lip} $\end{document} unimodal maps have unbounded topological entropy.

In this paper we study the following class of fractional relativistic Schrödinger equations:

\begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} (-\Delta+m^{2})^{s}u + V(\varepsilon x) u = f(u) &\text{ in } \mathbb{R}^{N}, \\ u\in H^{s}( \mathbb{R}^{N}), \quad u>0 &\text{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} $\end{document}

where \begin{document}$ \varepsilon >0 $\end{document} is a small parameter, \begin{document}$ s\in (0, 1) $\end{document}, \begin{document}$ m>0 $\end{document}, \begin{document}$ N> 2s $\end{document}, \begin{document}$ (-\Delta+m^{2})^{s} $\end{document} is the fractional relativistic Schrödinger operator, \begin{document}$ V: \mathbb{R}^{N} \rightarrow \mathbb{R} $\end{document} is a continuous potential satisfying a local condition, and \begin{document}$ f: \mathbb{R} \rightarrow \mathbb{R} $\end{document} is a continuous subcritical nonlinearity. By using a variant of the extension method and a penalization technique, we first prove that, for \begin{document}$ \varepsilon >0 $\end{document} small enough, the above problem admits a weak solution \begin{document}$ u_{\varepsilon } $\end{document} which concentrates around a local minimum point of \begin{document}$ V $\end{document} as \begin{document}$ \varepsilon \rightarrow 0 $\end{document}. We also show that \begin{document}$ u_{\varepsilon } $\end{document} has an exponential decay at infinity by constructing a suitable comparison function and by performing some refined estimates. Secondly, by combining the generalized Nehari manifold method and Ljusternik-Schnirelman theory, we relate the number of positive solutions with the topology of the set where the potential \begin{document}$ V $\end{document} attains its minimum value.

We prove global in time dispersion for the wave and the Klein-Gordon equation inside the Friedlander domain by taking full advantage of the space-time localization of caustics and a precise estimate of the number of waves that may cross at a given, large time. Moreover, we uncover a significant difference between Klein-Gordon and the wave equation in the low frequency, large time regime, where Klein-Gordon exhibits a worse decay than the wave, unlike in the flat space.

We study a periodic Kolmogorov system describing two species nonlinear competition. We discuss coexistence and extinction of one or both species, and describe the domain of attraction of nontrivial periodic solutions in the axes, under conditions that generalise Gopalsamy conditions. Finally, we apply our results to a model of microbial growth and to a model of phytoplankton competition under the effect of toxins.

In this note we study a new class of alignment models with self-propulsion and Rayleigh-type friction forces, which describes the collective behavior of agents with individual characteristic parameters. We describe the long time dynamics via a new method which allows us to reduce analysis from the multidimensional system to a simpler family of two-dimensional systems parametrized by a proper Grassmannian. With this method we demonstrate exponential alignment for a large (and sharp) class of initial velocity configurations confined to a sector of opening less than \begin{document}$ \pi $\end{document}.

In the case when characteristic parameters remain frozen, the system governs dynamics of opinions for a set of players with constant convictions. Viewed as a dynamical non-cooperative game, the system is shown to possess a unique stable Nash equilibrium, which represents a settlement of opinions most agreeable to all agents. Such an agreement is furthermore shown to be a global attractor for any set of initial opinions.

It is shown that an internal control based on a moving indicator function is able to stabilize the state of parabolic equations evolving in rectangular domains. For proving the stabilizability result, we start with a control obtained from an oblique projection feedback based on a finite number of static actuators, then we used the continuity of the state when the control varies in a relaxation metric to construct a switching control where at each given instant of time only one of the static actuators is active, finally we construct the moving control by traveling between the static actuators.

Numerical computations are performed by a concatenation procedure following a receding horizon control approach. They confirm the stabilizing performance of the moving control.

We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation

\begin{document}$ \begin{align*} \partial_{x}\left(u_{t}-\beta\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-\gamma u = 0 \end{align*} $\end{document}

in the anisotropic Sobolev spaces \begin{document}$ H^{s_{1},s_{2}}(\mathbb{R}^{2}) $\end{document}. When \begin{document}$ \beta <0 $\end{document} and \begin{document}$ \gamma >0, $\end{document} we prove that the Cauchy problem is locally well-posed in \begin{document}$ H^{s_{1}, s_{2}}(\mathbb{R}^{2}) $\end{document} with \begin{document}$ s_{1}>-\frac{1}{2} $\end{document} and \begin{document}$ s_{2}\geq 0 $\end{document}. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang(Transactions of the American Mathematical Society, 364(2012), 3395–3425.). The key idea is that we divide the frequency space into regular region and singular region. We further prove that the Cauchy problem for RMKP equation is ill-posed in \begin{document}$ H^{s_{1},0}(\mathbb{R}^{2}) $\end{document} with \begin{document}$ s_{1}<-\frac{1}{2} $\end{document} in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not \begin{document}$ C^{3} $\end{document}. When \begin{document}$ \beta <0,\gamma >0, $\end{document} by using the \begin{document}$ U^{p} $\end{document} and \begin{document}$ V^{p} $\end{document} spaces, we prove that the Cauchy problem is locally well-posed in \begin{document}$ H^{-\frac{1}{2},0}(\mathbb{R}^{2}) $\end{document}.

We propose and study a one-dimensional \begin{document}$ 2\times 2 $\end{document} hyperbolic Eulerian system with local relaxation from critical threshold phenomena perspective. The system features dynamic transition between strictly and weakly hyperbolic. For different classes of relaxation we identify intrinsic critical thresholds for initial data that distinguish global regularity and finite time blowup. For relaxation independent of density, we estimate bounds on density in terms of velocity where the system is strictly hyperbolic.

In this paper, we study discrete spectrum of invariant measures for countable discrete amenable group actions.

We show that an invariant measure has discrete spectrum if and only if it has bounded measure complexity. We also prove that, discrete spectrum can be characterized via measure-theoretic complexity using names of a partition and the Hamming distance, and it turns out to be equivalent to both mean equicontinuity and equicontinuity in the mean.

In [22], Jabłoński proved that a piecewise expanding \begin{document}$ C^{2} $\end{document} multidimensional Jabłoński map admits an absolutely continuous invariant probability measure (ACIP). In [6], Boyarsky and Lou extended this result to the case of i.i.d. compositions of the above maps, with an on average expanding condition. We generalize these results to the (quenched) setting of random Jabłoński maps, where the randomness is governed by an ergodic, invertible and measure preserving transformation. We prove that the skew product associated to this random dynamical system admits a finite number of ergodic ACIPs. Furthermore, we provide two different upper bounds on the number of mutually singular ergodic ACIPs, motivated by the works of Buzzi [9] in one dimension and Góra, Boyarsky and Proppe [19] in higher dimensions.

For any continuous self-map of a compact metric space, we define, prove the existence, and give an explicit expression of a maximal chain continuous factor. For the purpose, we exploit a chain proximal relation and its extension. An example is given to illustrate a difference of the two relations. An alternative proof of a result on the odometers and the regular recurrence is given. Also, we provide an example of a calculation of the maximal chain continuous factor for generic homeomorphism of the Cantor set.

The effect of decaying oscillatory perturbations on autonomous Hamiltonian systems in the plane with a stable equilibrium is investigated. It is assumed that perturbations preserve the equilibrium and satisfy a resonance condition. The behaviour of the perturbed trajectories in the vicinity of the equilibrium is investigated. Depending on the structure of the perturbations, various asymptotic regimes at infinity in time are possible. In particular, a phase locking and a phase drifting can occur in the systems. The paper investigates the bifurcations associated with a change of Lyapunov stability of the equilibrium in both regimes. The proposed stability analysis is based on a combination of the averaging method and the construction of Lyapunov functions.

We study the Mackey-Glass type monostable delayed reaction-diffusion equation with a unimodal birth function \begin{document}$ g(u) $\end{document}. This model, designed to describe evolution of single species populations, is considered here in the presence of the weak Allee effect (\begin{document}$ g(u_0)>g'(0)u_0 $\end{document} for some \begin{document}$ u_0>0 $\end{document}). We focus our attention on the existence of slow monotonic traveling fronts to the equation: under given assumptions, this problem seems to be rather difficult since the usual positivity and monotonicity arguments are not effective. First, we solve the front existence problem for small delays, \begin{document}$ h \in [0,h_p] $\end{document}, where \begin{document}$ h_p $\end{document}, given by an explicit formula, is optimal in a certain sense. Then we take a representative piece-wise linear unimodal birth function which makes possible explicit computation of traveling fronts. In this case, we find out that a) increase of delay can destroy asymptotically stable pushed fronts; b) the set of all admissible wavefront speeds has usual structure of a semi-infinite interval \begin{document}$ [c_*, +\infty) $\end{document}; c) for each \begin{document}$ h\geq 0 $\end{document}, the pushed wavefront is unique (if it exists); d) pushed wave can oscillate slowly around the positive equilibrium for sufficiently large delays.

This paper deals with the nutrient-taxis system derived by a food metric. The system was proposed in [Sun-Ho Choi and Yong-Jung Kim: Chemotactic traveling waves by metric of food, SIAM J. Appl. Math. 75 (2015), 2268–2289] using geometric ideas without gradient sensing, and has a simple form but contains a singular diffusive coefficient on the equation for the organism side. To overcome the difficulty arising from this singular structure, we use a weighted \begin{document}$ L^{p} $\end{document}-estimate involving a weighted Gagliardo-Nirenberg type inequality. In the one dimensional setting, it turns out that the system is shown to be globally well-posed in certain Sobolev spaces and the solutions are uniformly bounded. Moreover, the zero viscosity limit of the equation for the nutrient side is considered. For the same initial data and any given finite time interval, a diffusive solution converges to a non-diffusive solution when the diffusion coefficient vanishes.

We consider in this paper a diffusion-convection reaction equation in one space dimension. The main assumptions are about the reaction term, which is monostable, and the diffusivity, which changes sign once or even more than once; then, we deal with a forward-backward parabolic equation. Our main results concern the existence of globally defined traveling waves, which connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative. We also investigate the monotony of the profiles and show the appearance of sharp behaviors at the points where the diffusivity degenerates. In particular, if such points are interior points, then the sharp behaviors are new and unusual.