American Institute of Mathematical Sciences

In this paper, we study the existence and the nonexistence of positive classical solutions of the static Hartree-Poisson equation

\begin{document}$-Δ u = pu^{p-1}(|x|^{2-n}*u^p),\;\; u>0 \;\;in\;\; R^n, $ \end{document}

where \begin{document}$n ≥ 3$\end{document} and \begin{document}$p≥ 1$\end{document}. The exponents of the Serrin type, the Sobolev type and the Joseph-Lundgren type play the critical roles as in the study of the Lane-Emden equation. First, we prove that the equation has no positive solution when \begin{document}$1 ≤ p <\frac{n+2}{n-2}$\end{document} by means of the method of moving planes to the following system

\begin{document}$\left\{ \begin{array}{l} - \Delta u = \sqrt p {u^{p - 1}}v,\;\;u > 0\;\;in\;\;{R^n},\\ - \Delta v = \sqrt p {u^p},\;\;v > 0\;\;in\;\;{R^n}.\end{array} \right.$ \end{document}

When \begin{document}$p = \frac{n+2}{n-2}$\end{document}, all the positive solutions can be classified as

\begin{document}$u(x) = c(\frac{t}{t^2+|x-x^*|^2})^{\frac{n-2}{2}}$ \end{document}

with the help of an integral system involving the Newton potential, where \begin{document}$c, t$\end{document} are positive constants, and \begin{document}$x^* ∈ R^n$\end{document}. In addition, we also give other equivalent conditions to classify those positive solutions. When \begin{document}$p>\frac{n+2}{n-2}$\end{document}, by the shooting method and the Pohozaev identity, we find radial solutions for the system. In particular, the equation has a radial solution decaying with slow rate \begin{document}$\frac{2}{p-1}$\end{document}. Finally, we point out that the equation has positive stable solutions if and only if \begin{document}$p ≥ 1+\frac{4}{n-4-2\sqrt{n-1}}$\end{document}.

Consider the space of analytic, quasi-periodic cocycles on the higher dimensional torus. We provide examples of cocycles with nontrivial Lyapunov spectrum, whose homotopy classes do not contain any cocycles satisfying the dominated splitting property. This shows that the main result in the recent work "Complex one-frequency cocycles" by A. Avila, S. Jitomirskaya and C. Sadel does not hold in the higher dimensional torus setting.

In this paper, we study the following nonlinear Dirac equation

\begin{document}$\begin{equation*}-i\varepsilonα·\nabla w+aβ w+V(x)w = g(|w|)w, \ x∈ \mathbb{R}^3, \ {\rm for}\ w∈ H^1(\mathbb R^3, \mathbb C^4), \end{equation*}$ \end{document}

where \begin{document}$a > 0$\end{document} is a constant, \begin{document}$α = (α_1, α_2, α_3)$\end{document}, \begin{document}$α_1, α_2, α_3$\end{document} and \begin{document}$β$\end{document} are \begin{document}$4×4$\end{document} Pauli-Dirac matrices. Under the assumptions that \begin{document}$V$\end{document} and \begin{document}$g$\end{document} are continuous but are not necessarily of class \begin{document}$C^1$\end{document}, when \begin{document}$g$\end{document} is super-linear growth at infinity we obtain the existence of semiclassical solutions, which converge to the least energy solutions of its limit problem as \begin{document}$\varepsilon \to 0$\end{document}.

In this paper, we propose a two-component \begin{document}$b$\end{document}-family system with cubic nonlinearity and peaked solitons (peakons) solutions, which includes the celebrated Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation and its two-component extension as special cases. Firstly, we study single peakon and multi-peakon solutions to the system. Then the local well-posedness for the Cauchy problem of the system is discussed. Furthermore, we derive the precise blow-up scenario and global existence for strong solutions to the two-component \begin{document}$b$\end{document}-family system with cubic nonlinearity. Finally, we investigate the asymptotic behaviors of strong solutions at infinity within its lifespan provided the initial data decay exponentially and algebraically.

We examine the \begin{document} $N$ \end{document}-vortex problem on general domains \begin{document} $Ω\subset\mathbb{R}^2$ \end{document} concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system of the form

\begin{document}$Γ_k\dot{z}_k = J\nabla_{z_k}H(z_1,...,z_N),\ \ \ \ k = 1,...,N,$ \end{document}

where \begin{document} $Γ_k∈\mathbb{R}\setminus\{0\}$ \end{document} is the strength of the \begin{document} $k$ \end{document}th vortex at position \begin{document} $z_k(t)∈Ω$ \end{document}, \begin{document} $J∈\mathbb{R}^{2× 2}$ \end{document} is the standard symplectic matrix and

\begin{document}$H(z_1,...,z_N) = -\frac{1}{2π}\sum\limits_{\underset{k≠ j}{k,j = 1}}^NΓ_jΓ_k\log|z_k-z_j|-\sum\limits_{k,j = 1}^NΓ_jΓ_k g(z_k,z_j)$ \end{document}

with some regular and symmetric, but in general not explicitely known function \begin{document} $g:Ω×Ω \to \mathbb{R}$ \end{document}. The investigation relies on the idea to superpose a stationary solution of a system of less than \begin{document} $N$ \end{document} vortices and several clusters of vortices that are close to rigidly rotating configurations of the whole-plane system. We establish general conditions on both, the stationary solution and the configurations, under which multiple \begin{document} $T$ \end{document}-periodic solutions are shown to exist for every \begin{document} $T>0$ \end{document} small enough. The crucial condition holds in generic bounded domains and is explicitly verified for an example in the unit disc \begin{document} $Ω = B_1(0)$ \end{document}. In particular we therefore obtain various examples of periodic solutions in \begin{document} $B_1(0)$ \end{document} that are not rigidly rotating configurations.

In this paper, we study the following nonlinear Schrödinger-Poisson system

\begin{document}$\left\{\begin{array}{ll} -\Delta u+u+\epsilon K(x)\Phi(x)u = f(u),& x\in \mathbb{R}^{3} , \\ -\Delta \Phi = K(x)u^{2},\,\,& x\in \mathbb{R}^{3}, \\\end{array}\right.$ \end{document}

where \begin{document} $K(x)$ \end{document} is a positive and continuous potential and \begin{document} $f(u)$ \end{document} is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Under some suitable conditions, which are given in section 1, we prove that there exists some \begin{document} $\epsilon_{0}>0$ \end{document} such that for \begin{document} $0<\epsilon<\epsilon_{0}$ \end{document}, the above problem has infinitely many positive solutions by applying localized energy method. Our main result can be viewed as an extension to a recent result Theorem 1.1 of Ao and Wei in [3] and a result of Li, Peng and Wang in [26].

In this paper, we consider the orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity, which admits the single peakons and multi-peakons. We firstly show the existence of the single peakon and prove two useful conservation laws. Then by constructing certain Lyapunov functionals, we give the proof of stability result of peakons in the energy space \begin{document}$ H^1(\mathbb{R})$\end{document}-norm.

This paper is concerned with a two-component integrable Camassa-Holm type system with arbitrary smooth function \begin{document}$ H$\end{document}. If the function $H$ belongs to a set \begin{document}$ \mathcal{H}$\end{document} (defined in Section 4), then we obtain the existence and uniqueness of global strong solutions and global weak solutions to the system. Our obtained results generalize and improve considerably recent results in [38,39].

This article is concerned with the large data global regularity for the equivariant case of the classical Skyrme model and proves that this is valid for initial data in \begin{document}$H^s \times H^{s-1}(\mathbb{R}^3)$\end{document} with \begin{document}$s>7/2$\end{document}.

Let \begin{document}$X$\end{document} be a proper Hadamard space and \begin{document}$\Gamma <{\text{Is}}(X)$\end{document} a non-elementary discrete group of isometries with a rank one isometry. We discuss and prove Hopf-Tsuji-Sullivan dichotomy for the geodesic flow on the set of parametrized geodesics of the quotient \begin{document}$\Gamma \backslash X$\end{document} and with respect to Ricks' measure introduced in [35]. This generalizes previous work of the author and J. C. Picaud on Hopf-Tsuji-Sullivan dichotomy in the analogous manifold setting and with respect to Knieper's measure.

In this paper, we prove a well posedness result for an initial boundary value problem for a stochastic nonlocal reaction-diffusion equation with nonlinear diffusion together with a nul-flux boundary condition in an open bounded domain of \begin{document} $\mathbb{R}^n$ \end{document} with a smooth boundary. We suppose that the additive noise is induced by a Q-Brownian motion.

This work concerns the problem associated with averaging principle for a stochastic Kuramoto-Sivashinsky equation with slow and fast time-scales. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the stochastic Kuramoto-Sivashinsky equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single stochastic Kuramoto-Sivashinsky equation with a modified coefficient.

In this paper we investigate an alternating direction implicit (ADI) time integration scheme for the linear Maxwell equations with currents, charges and conductivity. We show its stability and efficiency. The main results establish that the scheme converges in a space similar to \begin{document}$H^{-1}$\end{document} with order two to the solution of the Maxwell system. Moreover, the divergence conditions in the system are preserved in \begin{document}$H^{-1}$\end{document} with order one.

Local correlation entropy, introduced by Takens in 1983, represents the exponential decay rate of the relative frequency of recurrences in the trajectory of a point, as the embedding dimension grows to infinity. In this paper we study relationship between the supremum of local correlation entropies and the topological entropy. For dynamical systems on topological graphs we prove that the two quantities coincide. Moreover, there is an uncountable set of points with local correlation entropy arbitrarily close to the topological entropy. On the other hand, we construct a strictly ergodic subshift with positive topological entropy having all local correlation entropies equal to zero. As a necessary tool, we derive an expected relationship between the local correlation entropies of a system and those of its iterates.

We prove a new generation result in $L^1$ for a large class of non-local operators with non-degenerate local terms. This class contains the operators appearing in Fokker-Planck or Kolmogorov forward equations associated with Lévy driven SDEs, i.e. the adjoint operators of the infinitesimal generators of these SDEs. As a byproduct, we also obtain a new elliptic regularity result of independent interest. The main novelty in this paper is that we can consider very general Lévy operators, including state-space depending coefficients with linear growth and general Lévy measures which can be singular and have fat tails.

We improve previous results on dispersion decay for 3D KleinGordon equation with generic potential. We develop a novel approach, which allows us to establish the decay in more strong norms and to weaken assumptions on the potential.

Given a nonlinear control system depending on two controls \begin{document}$u$\end{document} and \begin{document}$v$\end{document}, with dynamics affine in the (unbounded) derivative of \begin{document}$u$\end{document} and a closed target set \begin{document}$\mathcal{S}$\end{document} depending both on the state and on the control \begin{document}$u$\end{document}, we study the minimum time problem with a bound on the total variation of \begin{document}$u$\end{document} and \begin{document}$u$\end{document} constrained in a closed, convex set \begin{document}$U$\end{document}, possibly with empty interior. We revisit several concepts of generalized control and solution considered in the literature and show that they all lead to the same minimum time function \begin{document}$T$\end{document}. Then we obtain sufficient conditions for the existence of an optimal generalized trajectory-control pair and study the possibility of Lavrentiev-type gap between the minimum time in the spaces of regular (that is, absolutely continuous) and generalized controls. Finally, under a convexity assumption on the dynamics, we characterize \begin{document}$T$\end{document} as the unique lower semicontinuous solution of a regular HJ equation with degenerate state constraints.

In this paper, we consider nonlocal Schrödinger equations with certain potentials \begin{document}$V∈{\rm{RH}}^q$\end{document}(\begin{document}$q>\frac{n}{2s}>1$\end{document} and \begin{document}$0<s <1$\end{document}) of the form

\begin{document}$\begin{equation*}L_K u+V u = f\,\,\text{ in }\; \mathbb{R}^n \end{equation*}$ \end{document}

where \begin{document}$L_K$\end{document} is an integro-differential operator. We denote the solution of the above equation by \begin{document}$\mathcal{S}_V f: = u$\end{document}, which is called the inverse of the nonlocal Schrödinger operator \begin{document}$L_K+V$\end{document} with potential \begin{document}$V$\end{document}; that is, \begin{document}$\mathcal{S}_V = (L_K+V)^{-1}$\end{document}. Then we obtain an improved version of the weak Harnack inequality of nonnegative weak subsolutions of the nonlocal equation

\begin{document}$\begin{equation}\begin{cases}L_K u+V u = 0\,\,&\text{ in }\; \Omega,\\ u = g\,\,&\text{ in }\; \mathbb{R}^n\backslash\Omega, \;\;\;\;\;\;\;\;\; (1)\end{cases}\end{equation}$ \end{document}

where \begin{document}$g∈ H^s(\mathbb{R}^n)$\end{document} and \begin{document}$\Omega$\end{document} is a bounded open domain in \begin{document}$\mathbb{R}^n$\end{document} with Lipschitz boundary, and also get an improved decay of a fundamental solution \begin{document}$\mathfrak{e}_V$\end{document} for \begin{document}$L_K+V$\end{document}. Moreover, we obtain \begin{document}$L^p$\end{document} and \begin{document}$L^p-L^q$\end{document} mapping properties of the inverse \begin{document}$\mathcal{S}_V$\end{document} of the nonlocal Schrödinger operator \begin{document}$L_K+V$\end{document}.

We consider a class of parametric Schrödinger equations driven by the fractional \begin{document}$p$\end{document}-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. Using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter.

Let X be a two-sided subshift on a finite alphabet endowed with a mixing probability measure which is positive on all cylinders in X. We show that there exists an arbitrarily small finite overlapping union of shifted cylinders which intersects every orbit under the shift map.

We also show that for any proper subshift Y of X there exists a finite overlapping unions of shifted cylinders such that its survivor set contains Y (in particular, it can have entropy arbitrarily close to the entropy of X). Both results may be seen as somewhat counter-intuitive.

Finally, we apply these results to a certain class of hyperbolic algebraic automorphisms of a torus.

Translating soliton is a special solution for the mean curvature flow (MCF) and the parabolic rescaling model of type Ⅱ singularities for the MCF. By introducing an appropriate coordinate transformation, we first show that there exist complete helicoidal translating solitons for the MCF in \begin{document}$\mathbb{R}^{3}$\end{document} and we classify the profile curves and analyze their asymptotic behavior. We rediscover the helicoidal translating solitons for the MCF which are founded by Halldorsson [10]. Second, for the pinch zero we rediscover rotationally symmetric translating solitons in \begin{document}$\Bbb R^{n+1}$\end{document} and analyze the asymptotic behavior of the profile curves using a dynamical system. Clearly rotational hypersurfaces are foliated by spheres. We finally show that translating solitons foliated by spheres become rotationally symmetric translating solitons with the axis of revolution parallel to the translating direction. Hence, we obtain that any translating soliton foliated by spheres becomes either an n-dimensional translating paraboloid or a winglike translator.

We introduce a method to study the long-time behavior of solutions to damped wave equations, where the coefficients of the equations are space-time dependent. We show that solutions exhibit the diffusion phenomenon, connecting their asymptotic behaviors with the asymptotic behaviors of solutions to corresponding parabolic equations. Sharp decay estimates for solutions to damped wave equations are given, and decay estimates for derivatives of solutions are also discussed.

In a bounded domain \begin{document}$\Omega\subset\mathbb{R}^n$\end{document}, where \begin{document}$n\ge 3$\end{document}, we consider the quasilinear parabolic-parabolic Keller-Segel system

\begin{document}$\begin{equation*}\begin{cases}u_t = \nabla\cdot({D(u)\nabla u+u\nabla v}) \;\;\; &\text{in}\ \Omega\times(0,\infty)\\v_t = \Delta v-v+u &\text{in}\ \Omega\times(0,\infty)\end{cases}\end{equation*}$ \end{document}

with homogeneous Neumann boundary conditions. We will find that the condition \begin{document}$D(u)\geq Cu^{m-1}$\end{document} suffices to prove the uniqueness and global existence of solutions along with their boundedness if \begin{document}$D(0)>0$\end{document} and \begin{document}$m>1+\frac{(n-2)(n-1)}{n^2}$\end{document} which is a very different result from what we know about the same system with nonnegative sensitivity. In the case of degenerate diffusion (\begin{document}$D(0) = 0$\end{document}) and for the same parameters, locally bounded global weak solutions will be established.