American Institute of Mathematical Sciences

This essay is concerned with long-crested waves such as those arising in bore propagation. Such motions obtain on rivers when a surge of water invades an otherwise constantly flowing stretch and in the run-up of waves in the near-shore zone of large bodies of water. The dominating feature of the motion is that, in a standard \begin{document}$ xyz- $\end{document}coordinate system in which \begin{document}$ z $\end{document} increases in the direction opposite to which gravity acts and \begin{document}$ x $\end{document} increases in the principal direction of propagation, the depth of the fluid approaches a constant value \begin{document}$ h_0>0 $\end{document} as \begin{document}$ x \to +\infty $\end{document} and another value \begin{document}$ h_1>h_0 $\end{document} as \begin{document}$ x \to -\infty $\end{document}. In an earlier work, the authors developed theory for an idealized model for such waves based on a Boussinesq system of equations. The local well-posedness theory developed in that article applies to the sort of initial data arising in modeling bore propagation. However, well-posedness on the longer, Boussinesq time scale was not dealt with in the case of bore propagation, though such results were established for motions where \begin{document}$ h_1 = h_0 $\end{document}.

We argue that without a well-posedness theory at least on the Boussinesq time scale, such models for bore-propagation may not be of any practical use. The issue of well-posedness is complicated by the fact that the total energy of the idealized initial data is infinite.

The theory makes its way via the derivation of suitable approximations with which to compare the full solution. An interesting feature of the theory is the determination of dynamical boundary behavior that is not prescribed, but which the solution necessarily satisfies.

We consider backward doubly stochastic differential equations (BDSDEs in short) driven by a Brownian motion and an independent Poisson random measure. We give sufficient conditions for the existence and the uniqueness of solutions of equations with Lipschitz generator which is, first, standard and then depends on the values of a solution in the past. We also prove comparison theorem for reflected BDSDEs.

We consider a nonlinear system of partial differential equations which describes the dynamics of two types of cell densities with contact inhibition. After a change of variables the system turns out to be parabolic-hyperbolic and admits travelling wave solutions which solve a 3D dynamical system. Compared to the scalar Fisher-KPP equation, the structure of the travelling wave solutions is surprisingly rich and to unravel part of it is the aim of the present paper. In particular, we consider a parameter regime where the minimal wave velocity of the travelling wave solutions is negative. We show that there exists a branch of travelling wave solutions for any nonnegative wave velocity, which is not connected to the travelling wave solution with minimal wave velocity. The travelling wave solutions with nonnegative wave velocity are strictly positive, while the solution with minimal one is segregated in the sense that the product \begin{document}$ uv $\end{document} vanishes.

We introduce a new mathematical framework for the time discretization of evolution equations with memory. As a model, we focus on an abstract version of the equation

\begin{document}$ \partial_t u(t) - \int_0^\infty g(s) \Delta u(t-s)\, {{\rm{d}}} s = 0 $\end{document}

with Dirichlet boundary conditions, modeling hereditary heat conduction with Gurtin-Pipkin thermal law. Well-posedness and exponential stability of the discrete scheme are shown, as well as the convergence to the solutions of the continuous problem when the time-step parameter vanishes.

In the early 1980's, computers made it possible to observe that in complex dynamics, one often sees dynamical behavior reflected in parameter space and vice versa. This duality was first exploited by Douady, Hubbard and their students in early work on rational maps.

Here, we extend these ideas to transcendental functions.

In [16], it was shown that for the tangent family, \begin{document}$ \{ \lambda \tan z \} $\end{document}, the hyperbolic components meet at a parameter \begin{document}$ \lambda^* $\end{document} such that \begin{document}$ f_{ \lambda^*}^n( \lambda^*i) = \infty $\end{document} for some \begin{document}$ n $\end{document}. The behavior there reflects the dynamic behavior of \begin{document}$ \lambda^* \tan z $\end{document} at infinity. In Part 1. we show that this duality extends to a more general class of transcendental meromorphic functions \begin{document}$ \{f_{\lambda}\} $\end{document} for which infinity is not an asymptotic value. In particular, we show that in "dynamically natural" one-dimensional slices of parameter space, there are "hyperbolic-like" components \begin{document}$ \Omega $\end{document} with a unique distinguished boundary point such that for \begin{document}$ \lambda \in \Omega $\end{document}, the dynamics of \begin{document}$ f_\lambda $\end{document} reflect the behavior of \begin{document}$ f_\lambda $\end{document} at infinity. Our main result is that every parameter point \begin{document}$ \lambda $\end{document} in such a slice for which the iterate of the asymptotic value of \begin{document}$ f_\lambda $\end{document} is a pole is such a distinguished boundary point.

In the second part of the paper, we apply this result to the families \begin{document}$ \lambda \tan^p z^q $\end{document}, \begin{document}$ p, q \in \mathbb Z^+ $\end{document}, to prove that all hyperbolic components of period greater than \begin{document}$ 1 $\end{document} are bounded.

In this paper, we consider a multigroup SIRS epidemic model with standard incidence rates and Markovian switching. Firstly, we obtain sufficient conditions for extinction of the diseases. Then we establish sufficient conditions for persistence in the mean of the diseases. Moreover, in the case of persistence, we derive sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching.

In this paper, we develop an operator splitting scheme for the fractional kinetic Fokker-Planck equation (FKFPE). The scheme consists of two phases: a fractional diffusion phase and a kinetic transport phase. The first phase is solved exactly using the convolution operator while the second one is solved approximately using a variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. We prove the convergence of the scheme to a weak solution to FKFPE. As a by-product of our analysis, we also establish a variational formulation for a kinetic transport equation that is relevant in the second phase. Finally, we discuss some extensions of our analysis to more complex systems.

In this paper, we obtain some estimations of the saddle order which is the sole topological invariant of the non-integrable resonant saddles of planar polynomial vector fields of arbitrary degree \begin{document}$n$\end{document}. Firstly, we prove that, for any given resonance \begin{document}$p:-q$\end{document}, \begin{document}$(p, q)=1$\end{document}, and sufficiently big integer \begin{document}$n$\end{document}, the maximal saddle order can grow at least as rapidly as \begin{document}$n^2$\end{document}. Secondly, we show that there exists an integer \begin{document}$k_0$\end{document}, which grows at least as rapidly as \begin{document}$3n^2/2$\end{document}, such that \begin{document}$L_{k_0}$\end{document} does not belong to the ideal generated by the first \begin{document}$k_0-1$\end{document} saddle values \begin{document}$L_1, L_2, \cdots, L_{k_0-1}$\end{document}, where \begin{document}$L_{k}$\end{document} represents the \begin{document}$k$\end{document}-th saddle value of the given system. In particular, if \begin{document}$p=1$\end{document} (or \begin{document}$q=1$\end{document}), we obtain a sharper result that \begin{document}$k_0$\end{document} can grow at least as rapidly as \begin{document}$2 n^2$\end{document}. These results are valid for both cases of real and complex resonant saddles.

We consider product of expansive Markov maps on an interval with hole which is conjugate to a subshift of finite type. For certain class of maps, it is known that the escape rate into a given hole does not just depend on its size but also on its position in the state space. We illustrate this phenomenon for maps considered here. We compare the escape rate into a connected hole and a hole which is a union of holes with a certain property, but have same measure. This gives rise to some interesting combinatorial problems.

We prove existence of solutions for a class of nonhomogeneous elliptic system with asymmetric nonlinearities that are resonant at −∞ and superlinear at +∞. The proof is based on topological degree arguments. A priori bounds for the solutions are obtained by adapting the method of BrezisTurner.

The restricted three-body problem is an important subject that deals with significant issues referring to scientific fields of celestial mechanics, such as analyzing asteroid movement behavior and orbit designing for space probes. The 2-center problem is its simplified model. The goal of this paper is to show the existence of brake orbits, which means orbits whose velocities are zero at some times, under some particular conditions in the 2-center problem by using variational methods.

This paper deals with the stability of semi-wavefronts to the following delay non-local monostable equation: \begin{document}$\dot{v}(t,x) = \Delta v(t,x) - v(t,x) + \int_{\mathbb{R}^d}K(y)g(v(t-h,x-y))dy, x \in \mathbb{R}^d,\ t >0;$\end{document} where \begin{document}$h>0$\end{document} and \begin{document}$d\in\mathbb{Z}_+$\end{document}. We give two general results for \begin{document}$d\geq1$\end{document}: on the global stability of semi-wavefronts in \begin{document}$L^p$\end{document}-spaces with unbounded weights and the local stability of planar wavefronts in \begin{document}$L^p$\end{document}-spaces with bounded weights. We also give a global stability result for \begin{document}$d = 1$\end{document} which yields to the global stability in Sobolev spaces with bounded weights. Here \begin{document}$g$\end{document} is not assumed to be monotone and the kernel \begin{document}$K$\end{document} is not assumed to be symmetric, therefore non-monotone semi-wavefronts and backward semi-wavefronts appear for which we show their stability. In particular, the global stability of critical wavefronts is stated.

We construct multiple sign-changing solutions for the nonhomogeneous nonlocal equation

\begin{document}$ (-\Delta_{\Omega})^{s} u = \left| u\right| ^{\frac{4}{N-2s}}u +\varepsilon f(x)\quad \mbox{in }\Omega, $\end{document}

under zero Dirichlet boundary conditions in a bounded domain \begin{document}$ \Omega $\end{document} in \begin{document}$ \mathbb{R}^{N} $\end{document}, \begin{document}$ N>4s $\end{document}, \begin{document}$ s\in (0,1] $\end{document}, with \begin{document}$ f\in L^{\infty}(\Omega) $\end{document}, \begin{document}$ f\geq 0 $\end{document} and \begin{document}$ f\neq0 $\end{document}. Here, \begin{document}$ \varepsilon>0 $\end{document} is a small parameter, and \begin{document}$ (-\Delta_{\Omega})^{s} $\end{document} represents a type of nonlocal operator sometimes called the spectral fractional Laplacian. We show that the number of sign-changing solutions goes to infinity as \begin{document}$ \varepsilon\rightarrow 0 $\end{document} when it is assumed that \begin{document}$ \Omega $\end{document} and \begin{document}$ f $\end{document} have certain smoothness and possess certain symmetries, and we are also able to establish accurately the contribution of the nonhomogeneous term in the found solutions. Our proof relies on the Lyapunov-Schmidt reduction method.

The aim of this paper is to classify the positive solutions of the nonlocal critical equation:

\begin{document}$ - \Delta u = \left( {{I_\mu }*{u^{2_\mu ^*}}} \right){u^{2_\mu ^* - 1}},x \in {{\mathbb{R}}^N}$ \end{document}

where \begin{document}$ 0<\mu<N $\end{document}, if \begin{document}$ N = 3\ \hbox{or} \ 4 $\end{document} and \begin{document}$ 0<\mu\leq4 $\end{document} if \begin{document}$ N\geq5 $\end{document}, \begin{document}$ I_{\mu} $\end{document} is the Riesz potential defined by

\begin{document}${I_\mu }(x) = \frac{{\Gamma \left( {\frac{\mu }{2}} \right)}}{{\Gamma \left( {\frac{{N - \mu }}{2}} \right){\pi ^{\frac{N}{2}}}{2^{N - \mu }}|x{|^\mu }}}$ \end{document}

with \begin{document}$ \Gamma(s) = \int^{+\infty}_{0}x^{s-1}e^{-x}dx $\end{document}, \begin{document}$ s>0 $\end{document} and \begin{document}$ 2^{\ast}_{\mu} = \frac{2N-\mu}{N-2} $\end{document} is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We apply the moving plane method in integral forms to prove the symmetry and uniqueness of the positive solutions. Moreover, we also prove the nondegeneracy of the unique solutions for the equation when \begin{document}$ \mu $\end{document} close to \begin{document}$ N $\end{document}.

It is shown that the planar Schrödinger-Poisson system with a general nonlinear interaction function has a nontrivial solution of mountain-pass type and a ground state solution of Nehari-Pohozaev type. The conditions on the nonlinear functions are much weaker and flexible than previous ones, and new variational and analytic techniques are used in the proof.

We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those systems. In particular, we prove that for a residual set of points in any minimal system, the set of return times to any non-empty, open set contains arbitrarily long geometric progressions. Under the separate assumptions of total minimality and distality, we prove that return time sets have positive multiplicative upper Banach density along \begin{document}$ \mathbb{N} $\end{document} and along cosets of multiplicative subsemigroups of \begin{document}$ \mathbb{N} $\end{document}, respectively. The primary motivation for this work is the long-standing open question of whether or not syndetic subsets of the positive integers contain arbitrarily long geometric progressions; our main result is some evidence for an affirmative answer to this question.

We study nonlinear Schrödinger \begin{document}$ i\partial_tu-Hu = F(u) $\end{document} (NLSH) equation associated to harmonic oscillator \begin{document}$ H = -\Delta +|x|^2 $\end{document} in modulation spaces \begin{document}$ M^{p,q}. $\end{document} When \begin{document}$ F(u) = (|x|^{-\gamma}\ast |u|^2)u, $\end{document} we prove global well-posedness for (NLSH) in modulation spaces \begin{document}$ M^{p,p}(\mathbb R^d) $\end{document} \begin{document}$ (1\leq p < 2d/(d+\gamma), 0<\gamma< \min \{ 2, d/2\}). $\end{document} When \begin{document}$ F(u) = (K\ast |u|^{2k})u $\end{document} with \begin{document}$ K\in \mathcal{F}L^q $\end{document} (Fourier-Lebesgue spaces) or \begin{document}$ M^{\infty,1} $\end{document} (Sjöstrand's class) or \begin{document}$ M^{1, \infty}, $\end{document} some local and global well-posedness for (NLSH) are obtained in some modulation spaces. As a consequence, we can get local and global well-posedness for (NLSH) in a function spaces\begin{document}$ - $\end{document}which are larger than usual \begin{document}$ L^p_s- $\end{document}Sobolev spaces.

In this paper, we study traveling wave solutions of the chemotaxis system

\begin{document}$ \begin{matrix} \left\{ \begin{array}{*{35}{l}} {{u}_{t}} = \Delta u-{{\chi }_{1}}\nabla (u\nabla {{v}_{1}})+{{\chi }_{2}}\nabla (u\nabla {{v}_{2}})+u(a-bu),\qquad \ x\in \mathbb{R} \\ \tau {{\partial }_{t}}{{v}_{1}} = (\Delta -{{\lambda }_{1}}I){{v}_{1}}+{{\mu }_{1}}u,\qquad \ x\in \mathbb{R}, \\ \tau \partial {{v}_{2}} = (\Delta -{{\lambda }_{2}}I){{v}_{2}}+{{\mu }_{2}}u,\qquad \ \ x\in \mathbb{R}, \\\end{array} \right. & (0.1) \\\end{matrix} $\end{document}

where \begin{document}$ \tau>0,\chi_{i}> 0,\lambda_i> 0,\ \mu_i>0 $\end{document} (\begin{document}$ i = 1,2 $\end{document}) and \begin{document}$ \ a>0,\ b> 0 $\end{document} are constants. Under some appropriate conditions on the parameters, we show that there exist two positive constant \begin{document}$ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)<c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) $\end{document} such that for every \begin{document}$ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)\leq c<c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2, \lambda_2) $\end{document}, (0.1) has a traveling wave solution \begin{document}$ (u,v_1,v_2)(x,t) = (U,V_1,V_2)(x-ct) $\end{document} connecting \begin{document}$ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $\end{document} and \begin{document}$ (0,0,0) $\end{document} satisfying

\begin{document}$ \lim\limits_{z\to \infty}\frac{U(z)}{e^{-\mu z}} = 1, $\end{document}

where \begin{document}$ \mu\in (0,\sqrt a) $\end{document} is such that \begin{document}$ c = c_\mu: = \mu+\frac{a}{\mu} $\end{document}. Moreover,

\begin{document}$ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = \infty $\end{document}

and

\begin{document}$ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = c_{\tilde{\mu}^*}, $\end{document}

where \begin{document}$ \tilde{\mu}^* = {\min\{\sqrt{a}, \sqrt{\frac{\lambda_1+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_2+\tau a}{(1-\tau)_{+}}}\}} $\end{document}. We also show that (1) has no traveling wave solution connecting \begin{document}$ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $\end{document} and \begin{document}$ (0,0,0) $\end{document} with speed \begin{document}$ c<2\sqrt{a} $\end{document}.

The paper investigates the attractors and their robustness for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping. We prove that the related evolution process has a finite-dimensional pullback attractor \begin{document}$ \mathscr{A}_\kappa $\end{document} and a pullback exponential attractor \begin{document}$ \mathscr{M}^\kappa_{exp} $\end{document} for each extensibility parameter \begin{document}$ \kappa\in [0,1] $\end{document}, respectively, and both of them are stable on the perturbation \begin{document}$ \kappa $\end{document}. In particular, these stability holds for the global and exponential attractors when the non-autonomous dynamical system degenerates to an autonomous one, so the results of the paper deepen and extend those in recent literatures [22,33].

The article addresses the isomorphism problem for non-minimal topological dynamical systems with discrete spectrum, giving a solution under appropriate topological constraints. Moreover, it is shown that trivial systems, group rotations and their products, up to factors, make up all systems with discrete spectrum. These results are then translated into corresponding results for non-ergodic measure-preserving systems with discrete spectrum.

We investigate the blow-up phenomena for the two-component generalizations of Camassa-Holm equation on the real line. We establish some a local-in-space blow-up criterion for system of coupled equations under certain natural initial profiles. Presented result extends and specifies the earlier blow-up criteria for such type systems.

Based on the \begin{document}$ H^2 $\end{document} existence of the solution, we investigate weighted estimates for a mixed boundary elliptic system in a two-dimensional corner domain, when the contact angle \begin{document}$ \omega\in(0,\pi/2) $\end{document}. This system is closely related to the Dirichlet-Neumann operator in the water-waves problem, and the weight we choose is decided by singularities of the mixed boundary system. Meanwhile, we also prove similar weighted estimates with a different weight for the Dirichlet boundary problem as well as the Neumann boundary problem when \begin{document}$ \omega\in(0,\pi) $\end{document}.

In this paper, we consider the following coupled nonlinear Schrödinger system

\begin{document}$ \left\{\begin{array}{ll} -\Delta u+(1+\delta a(x))u = \mu_1 u^3+\beta uv^2 &\mbox{in }\mathbb{R}^3,\\ -\Delta v+(1+\delta b(x))v = \mu_2 v^3+\beta u^2v &\mbox{in }\mathbb{R}^3,\\ u\to 0,\quad v\to 0, &\mbox{as } |x|\to\infty \end{array}\right. $\end{document}

where \begin{document}$ \mu_1>0 $\end{document}, \begin{document}$ \mu_2>0 $\end{document}, \begin{document}$ \beta\in\mathbb{R} $\end{document}, \begin{document}$ \delta\in\mathbb{R} $\end{document}, and \begin{document}$ a(x) $\end{document} and \begin{document}$ b(x) $\end{document} are two \begin{document}$ C^\alpha $\end{document} potentials with \begin{document}$ 0<\alpha<1 $\end{document}, satisfying some slow decay assumptions, but do not need to fulfill any symmetry property. Using the Lyapunov–Schmidt reduction method and some variational techniques, we show that there exist \begin{document}$ 0<\delta_0<1 $\end{document} and \begin{document}$ 0<\beta_0<\min\{\mu_1,\mu_2\} $\end{document} such that the above system has infinitely many positive segregated solutions for any \begin{document}$ 0<\delta<\delta_0 $\end{document} and \begin{document}$ 0<\beta<\beta_0 $\end{document}.

In this paper we consider the following Schrödinger-Poisson system with general nonlinearity

\begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\varepsilon^2\Delta u+V(x)u+\psi u = f(u),\,\, x\in\mathbb{R}^3,\\ -\varepsilon^2\Delta\psi = u^2,\,\,u>0,\,\, u\in H^1(\mathbb{R}^3),\\ \end{array} \right. \end{eqnarray*} $\end{document}

where \begin{document}$ \varepsilon>0 $\end{document} is a small positive parameter. Under a local condition imposed on the potential \begin{document}$ V $\end{document} and general conditions on \begin{document}$ f, $\end{document} we construct a family of positive semiclassical solutions. Moreover, the concentration phenomena around local minimum of \begin{document}$ V $\end{document} and exponential decay of semiclassical solutions are also explored. We do not need the monotonicity of the function \begin{document}$ u\rightarrow\frac{f(u)}{u^3} $\end{document}, and our results include the case \begin{document}$ f(u) = |u|^{p-2}u $\end{document} for \begin{document}$ 3<p<6 $\end{document}. Since without more global information on the potential, in the proofs we apply variational methods, penalization techniques and some analytical techniques.

A wide class of nonlinear dispersive wave equations are shown to possess a novel type of peakon solution in which the amplitude and speed of the peakon are time-dependent. These novel dynamical peakons exhibit a wide variety of different behaviours for their amplitude, speed, and acceleration, including an oscillatory amplitude and constant speed which describes a peakon breather. Examples are presented of families of nonlinear dispersive wave equations that illustrate various interesting behaviours, such as asymptotic travelling-wave peakons, dissipating/anti-dissipating peakons, direction-reversing peakons, runaway and blow up peakons, among others.

We study the asymptotic and qualitative properties of least energy radial sign-changing solutions to fractional semilinear elliptic problems of the form

\begin{document}$ \begin{cases} (-\Delta)^s u = |u|^{2^*_s-2-\varepsilon}u &\text{in } B_R, \\ u = 0 &\text{in } \mathbb{R}^n \setminus B_R, \end{cases} $\end{document}

where \begin{document}$ s \in (0,1) $\end{document}, \begin{document}$ (-\Delta)^s $\end{document} is the s-Laplacian, \begin{document}$ B_R $\end{document} is a ball of \begin{document}$ \mathbb{R}^n $\end{document}, \begin{document}$ 2^*_s : = \frac{2n}{n-2s} $\end{document} is the critical Sobolev exponent and \begin{document}$ \varepsilon>0 $\end{document} is a small parameter. We prove that such solutions have the limit profile of a "tower of bubbles", as \begin{document}$ \varepsilon \to 0^+ $\end{document}, i.e. the positive and negative parts concentrate at the same point with different concentration speeds. Moreover, we provide information about the nodal set of these solutions.