American Institute of Mathematical Sciences

Let \begin{document}$ u $\end{document} be a nonnegative solution to the equation

\begin{document}$ (-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * |x|^a u^p \right) |x|^a u^{p-1} \quad\text{ in } \mathbb{R}^n \setminus \{0\}, $\end{document}

where \begin{document}$ n \ge 2 $\end{document}, \begin{document}$ 0 < \alpha < 2 $\end{document}, \begin{document}$ 0 < \beta < n $\end{document} and \begin{document}$ a>\max\{ -\alpha, -\frac{\alpha+\beta}{2} \} $\end{document}. By exploiting the method of scaling spheres and moving planes in integral forms, we show that \begin{document}$ u $\end{document} must be zero if \begin{document}$ 1\le p<\frac{n+\beta+2a}{n-\alpha} $\end{document} and must be radially symmetric about the origin if \begin{document}$ a<0 $\end{document} and \begin{document}$ \frac{n+\beta+2a}{n-\alpha} \le p \le \frac{n+\beta+a}{n-\alpha} $\end{document}.

Let \begin{document}$ X $\end{document} be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. We prove the Bowen-Margulis measure on the space of geodesics is the unique measure of maximal entropy for the geodesic flow, which has topological entropy equal to the critical exponent of the Poincaré series.

We construct a mean Li-Yorke chaotic set along polynomial sequences (the degree of this polynomial is not less than three) with full Hausdorff dimension and full topological entropy for \begin{document}$ \beta $\end{document}-transformation. An uncountable subset \begin{document}$ C $\end{document} is said to be a mean Li-Yorke chaotic set along sequence \begin{document}$ \{a_n\} $\end{document}, if both

\begin{document}$ \begin{equation*} \liminf\limits_{N\to \infty}\frac{1}{N}\sum\limits_{j = 1}^{N}d(f^{a_j}(x),f^{a_j}(y )) = 0 \text{ and } \limsup\limits_{N\to \infty}\frac{1}{N}\sum\limits_{j = 1}^{N}d(f^{a_j}(x),f^{a_j}(y ))>0 \end{equation*} $\end{document}

hold for any two distinct points \begin{document}$ x $\end{document} and \begin{document}$ y $\end{document} in \begin{document}$ C $\end{document}.

In this paper, we first establish a narrow region principle for systems involving the fractional Laplacian in unbounded domains, which plays an important role in carrying on the direct method of moving planes. Then combining this direct method with the sliding method, we derive the monotonicity of bounded positive solutions to the following fractional Laplacian systems in unbounded Lipschitz domains \begin{document}$ \Omega $\end{document}

\begin{document}$ \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} \left(-\Delta\right)^{s}u& = &f(u,v), & \mbox{in}\ \ \Omega\,, \\[0.05cm] \left(-\Delta\right)^{t}v& = &g(u,v),& \mbox{in}\ \ \Omega\,, \\[0.05cm] u,\,v&\equiv&0, & \mbox{on}\ \ \mathbb{R}^{n}\setminus\Omega\,, \end{array}\right. \end{equation*} $\end{document}

without any decay assumptions on the solution pair \begin{document}$ (u,\,v) $\end{document} at infinity.

In this paper, we study a local rigidity property of \begin{document}$ \mathbb Z \ltimes_\lambda \mathbb R $\end{document} affine action on tori generated by an irreducible toral automorphism and a linear flow along an eigenspace. Such an action exhibits a weak version of local rigidity, i.e., any smooth perturbations close enough to an affine action is smoothly conjugate to the affine action up to constant time change.

In this paper, the non-autonomous dynamical behavior of weakly damped wave equation with a sup-cubic nonlinearity is considered in locally uniform spaces. We first prove the global well-posedness of the Shatah-Struwe solutions, then establish the existence of the \begin{document}$ \big(H_{lu}^{1}(\mathbb{R}^{3})\times L_{lu}^{2}(\mathbb{R}^{3}),H_{\rho}^{1}(\mathbb{R}^{3})\times L_{\rho}^{2}(\mathbb{R}^{3})\big) $\end{document}-pullback attractor for the Shatah-Struwe solutions process of this equation. The results are based on the recent extension of Strichartz estimates for the bounded domains.

For a dynamical system \begin{document}$ (X, T) $\end{document}, \begin{document}$ l\in\mathbb{N} $\end{document} and \begin{document}$ x\in X $\end{document}, let \begin{document}$ \mathbf{Q}^{[l]}(X) $\end{document} and \begin{document}$ \overline{\mathcal{F}^{[l]}}(x^{[l]}) $\end{document} be the orbit closures of the diagonal point \begin{document}$ x^{[l]} $\end{document} under the parallelepipeds group \begin{document}$ \mathcal{G}^{[l]} $\end{document} and the face group \begin{document}$ \mathcal{F}^{[l]} $\end{document} actions respectively. In this paper, it is shown that for a minimal system \begin{document}$ (X, T) $\end{document} and every \begin{document}$ l\in \mathbb{N}, x\in X $\end{document}, the maximal factors of order \begin{document}$ d $\end{document} of \begin{document}$ (\mathbf{Q}^{[l]}(X), \mathcal{G}^{[l]}) $\end{document} and \begin{document}$ (\overline{\mathcal{F}^{[l]}}(x^{[l]}), \mathcal{F}^{[l]}) $\end{document} are \begin{document}$ (\mathbf{Q}^{[l]}(X_d), \mathcal{G}^{[l]}) $\end{document} and \begin{document}$ (\overline{\mathcal{F}^{[l]}}(\pi(x)^{[l]}), \mathcal{F}^{[l]}) $\end{document} respectively, where \begin{document}$ \pi:X\to X/\mathbf{RP}^{[d]}(X) = X_d, d\in \mathbb{N}\cup\{\infty\} $\end{document} is the factor map and \begin{document}$ \mathbf{RP}^{[d]}(X) $\end{document} is the regionally proximal relation of order \begin{document}$ d $\end{document}.

In this work we consider the existence of positive singular solutions

\begin{document}$ \begin{equation} \left\{ \begin{array}{lcl} \hfill -\Delta u_1 & = & \lambda_1 | \nabla u_2|^p \qquad \mbox{ in } \Omega, \\ \hfill -\Delta u_2 & = & \lambda_2 | \nabla u_1|^q \qquad \mbox{ in } \Omega, \\ \hfill u_1 = u_2 & = & 0 \hfill \mbox{ on } \partial \Omega, \end{array}\right.\;\;\;\;\;\;\;(1) \end{equation} $\end{document}

where \begin{document}$ \Omega $\end{document} is small \begin{document}$ C^2 $\end{document} perturbation of the unit ball \begin{document}$ B_1 $\end{document} in \begin{document}$ \mathbb{R}^N $\end{document} and \begin{document}$ \lambda_i $\end{document} are positive constants. Under suitable conditions on \begin{document}$ p $\end{document} and \begin{document}$ q $\end{document} we prove the existence of positive singular solutions of (1). We also examine the case where one or both of \begin{document}$ u_1,u_2 $\end{document} are Hölder continuous.

This paper is concerned with the uniform stability estimate to the Cauchy problem of the Vlasov-Poisson-Boltzmann system. Our analysis is based on compensating function introduced by Kawashima and the standard energy method.

We consider the following Dirichlet problem

\begin{document}$\left\{ \begin{matrix} -\Delta u=\lambda f(u)\ \ \ \ \text{in}\ \Omega \\ u=0\ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial \Omega \\\end{matrix} \right.,\ \ \ \ \ \ \ \left( \mathcal{P}_{f}^{\lambda } \right)$ \end{document}

with \begin{document}$ \lambda<0 $\end{document} and \begin{document}$ f $\end{document} non-negative and non-decreasing.

We show existence and uniqueness of solutions \begin{document}$ u_\lambda $\end{document} for any \begin{document}$ \lambda $\end{document} and discuss their asymptotic behavior as \begin{document}$ \lambda\to-\infty $\end{document}. In the expansion of \begin{document}$ u_\lambda $\end{document} large solutions naturally appear.

We consider the focusing \begin{document}$ L^2 $\end{document}-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle \begin{document}$ \Theta \subset \mathbb{R}^3 $\end{document}. We construct a solution behaving asymptotically as a solitary wave on \begin{document}$ \mathbb{R}^3, $\end{document} for large times. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solutions of the NLS equation blowing up at several points together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at the soliton. These solutions are arbitrarily close to the scattering threshold given by a previous work of R. Killip, M. Visan and X. Zhang, which is the same as the one on the whole Euclidean space given by S. Roundenko and J. Holmer in the radial case and by the previous authors with T. Duyckaerts in the non-radial case.

Exchange-driven growth (EDG) is a model in which pairs of clusters interact by exchanging single unit with a rate given by a kernel \begin{document}$ K(j, k) $\end{document}. Despite EDG model's common use in the applied sciences, its rigorous mathematical treatment is very recent. In this article we study the large time behaviour of EDG equations. We show two sets of results depending on the properties of the kernel \begin{document}$ (i) $\end{document} \begin{document}$ K(j, k) = b_{j}a_{k} $\end{document} and \begin{document}$ (ii) $\end{document} \begin{document}$ K(j, k) = ja_{k}+b_{j}+\varepsilon\beta_{j}\alpha_{k} $\end{document}. For type I kernels, under the detailed balance assumption, we show that the system admits unique equilibrium up to a critical mass \begin{document}$ \rho_{s} $\end{document} above which there is no equilibrium. We prove that if the system has an initial mass below \begin{document}$ \rho_{s} $\end{document} then the solutions converge to a unique equilibrium distribution strongly where if the initial mass is above \begin{document}$ \rho_{s} $\end{document} then the solutions converge to cricital equilibrium distribution in a weak sense. For type II kernels, we do not make any assumption of detailed balance and equilibrium is shown as a consequence of contraction properties of solutions. We provide two separate results depending on the monotonicity of the kernel or smallness of the total mass. For the first case we prove exponential convergence in the number of clusters norm and for the second we prove exponential convergence in the total mass norm.

In this paper we establish an attainability result for the minimum time function of a control problem in the space of probability measures endowed with Wasserstein distance. The dynamics is provided by a suitable controlled continuity equation, where we impose a nonlocal nonholonomic constraint on the driving vector field, which is assumed to be a Borel selection of a given set-valued map. This model can be used to describe at a macroscopic level a so-called multiagent system made of several possible interacting agents.

This paper is devoted to the analytical study of the long-time asymptotic behavior of solutions to the Cauchy problem of a system of conservation laws in one space dimension, which is derived from a repulsive chemotaxis model with singular sensitivity and nonlinear chemical production rate. Assuming the \begin{document}$ H^2 $\end{document}-norm of the initial perturbation around a constant ground state is finite and using energy methods, we show that there exists a unique global-in-time solution to the Cauchy problem, and the constant ground state is globally asymptotically stable. In addition, the explicit decay rates of the solutions to the chemically diffusive and non-diffusive models are identified under different exponent ranges of the nonlinear chemical production function.

We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric \begin{document}$ \alpha $\end{document}-stable Lévy processes with values in \begin{document}$ {{{\mathbb R}}}^d $\end{document} under some mild Hölder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index \begin{document}$ \alpha \in (0,1) $\end{document}. New strong well-posedness results are also derived from the previous analysis.

A recent refinement of Kerékjártó's Theorem has shown that in \begin{document}$ \mathbb R $\end{document} and \begin{document}$ \mathbb R^2 $\end{document} all \begin{document}$ \mathcal C^l $\end{document}–solutions of the functional equation \begin{document}$ f^n = \text{Id} $\end{document} are \begin{document}$ \mathcal C^l $\end{document}–linearizable, where \begin{document}$ l\in \{0,1,\dots \infty\} $\end{document}. When \begin{document}$ l\geq 1 $\end{document}, in the real line we prove that the same result holds for solutions of \begin{document}$ f^n = f $\end{document}, while we can only get a local version of it in the plane. Through examples, we show that these results are no longer true when \begin{document}$ l = 0 $\end{document} or when considering the functional equation \begin{document}$ f^n = f^k $\end{document} with \begin{document}$ n>k\geq 2 $\end{document}.

In this paper we study the incompressible limit of the compressible inertial Qian-Sheng model for liquid crystal flow. We first derive the uniform energy estimates on the Mach number \begin{document}$ \epsilon $\end{document} for both the compressible system and its differential system with respect to time under uniformly in \begin{document}$ \epsilon $\end{document} small initial data. Then, based on these uniform estimates, we pass to the limit in the compressible system as \begin{document}$ \epsilon \rightarrow 0 $\end{document}, so that we establish the global classical solution of the incompressible system by compactness arguments. We emphasize that, on global in time existence of the incompressible inertial Qian-Sheng model under small size of initial data, the range of our assumptions on the coefficients are significantly enlarged, comparing to the results of De Anna and Zarnescu's work [6]. Moreover, we also obtain the convergence rates associated with \begin{document}$ L^2 $\end{document}-norm with well-prepared initial data.

In this paper we consider the Degasperis-Procesi equation, which is an approximation to the incompressible Euler equation in shallow water regime. First we provide the existence of solitary wave solutions for the original DP equation and the general theory of geometric singular perturbation. Then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Degasperis-Procesi equation is transformed into the slow-fast system by using the traveling wave transformation. It is proved that the perturbed equation also has a homoclinic orbit, which corresponds to a solitary wave solution of the delayed Degasperis-Procesi equation.

In this paper, we prove the existence of a nontrivial solution for the following boundary value problem

1 \begin{document}$ \left\{ {\begin{array}{*{20}{l}}{ - {\rm{div}}(\omega (x)|\nabla u(x){|^{N - 2}}\nabla u(x)) = f(x,u),\;\;\quad }&{\;\;\;\;\;{\rm{in}}\;B;}\\{u = 0,\;\;\quad }&{\;\;\;\;\;{\rm{on}}\;\partial B,}\end{array}} \right.{\rm{ }}\;\;\;\;\;\;\;\left( 1 \right)$\end{document}

where \begin{document}$ B $\end{document} is the unit ball in \begin{document}$ \mathbb{R}^N $\end{document}, \begin{document}$ N\geq 2 $\end{document}, the radial positive weight \begin{document}$ \omega(x) $\end{document} is of logarithmic type, the function \begin{document}$ f(x,u) $\end{document} is continuous in \begin{document}$ B\times\mathbb{R} $\end{document} and has critical double exponential growth, which behaves like \begin{document}$ \exp\{e^{\alpha |u|^{\frac{N}{N-1}}}\} $\end{document} as \begin{document}$ |u|\to\infty $\end{document} for some \begin{document}$ \alpha>0 $\end{document}.

We study internal Lie algebras in the category of subshifts on a fixed group – or Lie algebraic subshifts for short. We show that if the acting group is virtually polycyclic and the underlying vector space has dense homoclinic points, such subshifts can be recoded to have a cellwise Lie bracket. On the other hand there exist Lie algebraic subshifts (on any finitely-generated non-torsion group) with cellwise vector space operations whose bracket cannot be recoded to be cellwise. We also show that one-dimensional full vector shifts with cellwise vector space operations can support infinitely many compatible Lie brackets even up to automorphisms of the underlying vector shift, and we state the classification problem of such brackets.

From attempts to generalize these results to other acting groups, the following questions arise: Does every f.g. group admit a linear cellular automaton of infinite order? Which groups admit abelian group shifts whose homoclinic group is not generated by finitely many orbits? For the first question, we show that the Grigorchuk group admits such a CA, and for the second we show that the lamplighter group admits such group shifts.