We consider the Cauchy problem for the cubic nonlinear Schrödinger equation \begin{equation} \begin{cases} iu_{t}+\frac{1}{2}u_{xx}=u^{3},\text{ }x \in \mathbf{R},\text{ }t>0, \\ u(0,x)=u_{0}(x),\text{ }x\in \mathbf{R.} \end{cases} \label{A} \end{equation} The aim of the present paper is to consider problem (0.1) in low-order Sobolev spaces, when the initial data $u_{0}\in \mathbf{H}^{\alpha }\cap \mathbf{H}^{0,\alpha }$ with $\alpha >\frac{1}{2}.$ In our previous paper [7] we proved the global existence of solutions to (0.1) if the initial data $u_{0}\in \mathbf{H}^{2}\cap \mathbf{H}^{0,2}$. Also we find the large-time asymptotics of solutions.

In this paper we establish the existence of positive solutions for two kinds of nonlinear differential systems.

Using a fixed--point theorem of cone expansion/compression type, we show the existence of at least three positive radial solutions for the class of quasi-- linear elliptic systems \begin{equation*} \left\{ \begin{array}{rclcl} -\Delta_p u & = & \lambda k_1(|x|) f(u,v) & \mbox{ in } & \Omega, \\ -\Delta_q v & = & \lambda k_2(|x|) g(u,v) & \mbox{ in } & \Omega, \\ ( u,v) & = & (a,b) & \mbox{ on } & \partial \Omega \end{array} \right. \end{equation*} where the nonlinearities $ f, g \in C([0, +\infty)^2; [0,+\infty)) $ are superlinear at zero and sublinear at $ + \infty. $ The parameters $\lambda, a $ and $ b $ are positive, $ \Omega $ is the ball in $\mathbb{R}^N$, with $ N \ge 3, $ of radius $ R_0 $ which is centered at the origin, $ 1< p, \; q\le 2$, and $ k_1, k_2 \in C ([0, R_0]; [0, +\infty)).$

We study positive radial solutions to the problem \begin{equation} \label{maineqn1} \begin{cases} -\Delta u = \lambda K(\left|x \right|) f(u) \quad & x \in \Omega, \\ \hskip 15pt u = 0 & \mbox {if } \left|x \right| = r_0, \\ \hskip 15pt u \rightarrow0 & \mbox{as } \left|x \right|\rightarrow \infty , \end{cases} \end{equation} where $\lambda$ is a positive parameter, $\Delta u=\mbox{div} \big(\nabla u\big)$ is the Laplacian of $u$, $\Omega=\{x\in\ \mathbb{R} ^{n},n>2 : \left|x \right|>r_0\}$ is an exterior domain and $f:(0,\infty)\rightarrow \mathbb{R}$ belongs to a class of sublinear functions at $\infty$ such that they are continuous and $f(0^+)= \lim_{s \to 0^+} f(s) < 0$. In particular we also study infinite semipositone problems where $\lim_{s\rightarrow 0^{+}} f(s) =-\infty$. Here $K: [r_0,\infty)\rightarrow(0,\infty)$ belongs to a class of continuous functions such that $\lim_{r\rightarrow \infty}K(r)=0$. We establish various existence results for such boundary value problems and also extend our results to classes of systems. We prove our results by the method of sub-/supersolutions.

This paper is devoted to a systematic study of quadratic systems of second-order ordinary differential equations which are defined in commutative algebras. Conserved quantities are found and several examples defined in Jordan and Nahm algebras are analyzed. Hamiltonian and gradient systems are considered.

In this manuscript, the existence of traveling-wave solutions to Boussinesq systems \begin{equation*} \left\{ \begin{matrix} \eta_t +u_x+(\eta u)_x +au_{xxx}-b \eta_{xxt}=0, &\\ u_t + \eta_x +uu_x +c\eta_{xxx} - d u_{xxt}=0,&\\ \end{matrix} \right. \end{equation*} is established. We prove that all the systems with $ a<0,$ $ c<0$ and $ b=d$ exhibit traveling-wave solutions with small propagation speeds. The result complements our earlier work [6] on a restricted family of the systems where both existence and stability of traveling-wave solutions were established in the presence of large surface tension, namely when $a+b+c+d<0$.

We consider the behavior of nonnegative solutions to the Cauchy problem of the porous medium equation with localized reaction term: \begin{eqnarray*} \left\{ \begin{array}{ll} u_t = \Delta(u^m) + a(x)u^p, & (x,t) \in \mathbf{R}^n \times (0,T),\\ u(x,0) = u_0(x), & x \in \mathbf{R}^n, \end{array} \right. \end{eqnarray*} where $ m > 1 $, $ p > 0 $, $ a(x) \geq 0 $ is a compactly supported function, and $u_0(x)$ is continuous, nonnegative but not identical with zero, and has compact support as well. We show the relationship between the occurrence of blow-ups and the exponents $m$ and $p $: in two-dimensional space, all the solutions are globally defined if $0 < p \leq \frac{m+1}{2}$, and the solutions may blow up in finite time if $p \geq m$; in spaces higher than two-dimensional, all the solutions are global if $0 < p < m$, and there exist both global solutions and blow-up solutions if $p \geq m$. We also show that, for any solution, the intersection of its support and the support of $a(x)$ will be non-empty at some time.

We present a new proof of global existence and long range scattering, from small initial data, for the one--dimensional cubic gauge invariant nonlinear Schrödinger equation, and for Hartree equations in dimension $n \geq 2$. The proof relies on an analysis in Fourier space, related to the recent works of Germain, Masmoudi, and Shatah on space-time resonances. An interesting feature of our approach is that we are able to identify the long range phase correction term through a very natural stationary phase argument.

In this work, we consider a coupled system of wave-plate type with thermal effect. We show that the associated C$_0$-semigroup is analytic and exponentially stable.

We analyze the resolvent and define the scattering matrix for asymptotically hyperbolic manifolds with metrics which have a polyhomogeneous expansion near the boundary. We prove that there is always an essential singularity of the resolvent in this setting. We use this analysis to prove an inverse result for conformally compact odd-dimensional Einstein manifolds.