In this paper we study further properties of bounded continuous functions which behave like almost automorphic functions at infinity. Then we establish the existence and uniqueness of solutions of some semilinear functional differential and fractional differential equations in Banach spaces.

We consider a system of nonlinear Schrödinger equations with quadratic nonlinearities in two space dimensions. Our aim is to show time decay estimates of small solutions and nonexistence of the usual scattering states for a system. Furthermore we prove stability in time of small solutions in the neighborhood of solutions to a suitable approximate equation.

We study two boundary-value problems originally considered by A.C. Lazer and coauthors. The first is an elliptic problem \[ Lu+\alpha u+g \left( u\right) =h\left( x\right) \] in a bounded domain $\Omega\subset R^{N},$ with $u=0$ on the boundary $\partial\Omega.$ It is assumed that $Lu+\alpha u=0$ has a one-dimensional set of solutions satisfying the same boundary condition$.$ The second is an ODE problem \[ u^{\prime\prime}+n^{2}u+g\left( u\right) =e\left( t \right) , \] where $e$ has period $2\pi$ and a $2\pi$-periodic solution is sought. Here, the corresponding linear homogeneous equation, $u^{\prime\prime}+n^{2}u=0,$ has a two-dimensional set of $2\pi$-periodic solutions. In each case, conditions are sought which guarantee the existence of at least one solution to the original problem. We give short proofs of theorems first proved by E.M. Landesman and Lazer, and by Lazer and D.E. Leach, on these two problems.

In this paper we prove a general Strichartz estimate for certain perturbed wave equations, and here we can drop the nontrapping hypothesis and handle trapping obstacles with some loss of derivatives for data in the local energy decay estimates. We then give the obstacle version of the sharp life span for semilinear wave equations when $n=3,p<p_c$, by using the real interpolation method, and by getting corresponding finite time Strichartz estimates (see Section 3). Finally, as another application, we get the Strauss conjecture for semilinear wave equations with several convex obstacles when $n=3,4$ (see Section 4).

A 2-D incompressible channel flow with periodic condition along one axis is considered. The stabilization of it is achieved by a boundary feedback controller with tangential velocity observation which acts on the steamwise component of the velocity only. There are no a priori conditions imposed on the viscosity coefficient $ \nu$, that is, on the Reynolds number.

We show the following symmetry property of a bounded Reinhardt domain $\Omega$ in $\mathbb{C}^{n+1}$: let $M=\partial\Omega$ be the smooth boundary of $\Omega $ and let $h$ be the Second Fundamental Form of $M$; if the coefficient $h(T,T)$ related to the characteristic direction $T$ is constant then $M$ is a sphere. In the Appendix we state the result from a hamiltonian point of view.

Given $p$ points in a bounded domain of $\mathbb R^d$, with $d=2,3$, we show the existence of solutions of the $L^2$-critical focusing nonlinear Schrödinger equation blowing up exactly at these points.

The Keel-Smith-Sogge estimates are considered in a general frame in terms of the harmonic analysis. The estimates are applied to the Cauchy problem for nonlinear higher-order wave equations, and the existence of global and almost-global solutions is shown.

We investigate the initial-value problem (IVP) associated with the equation \begin{equation*} u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^5) =0, \end{equation*} where $g$ is a periodic function. We prove that, for given initial data $\phi \in H^1(\mathbb R)$, as $|\omega|\to \infty$, the solution $u_{\omega}$ converges to the solution $U$ of the initial-value problem associated with \begin{equation*} U_{t}+\partial_x^3U+m(g)\partial_x(U^5) =0, \end{equation*} with the same initial data, where $m(g)$ is the average of the periodic function $g$. Moreover, if the solution $U$ is global and satisfies $\|U\|_{L_x^5L_t^{10}}<\infty$, then we prove that the solution $u_{\omega}$ is also global provided $|\omega|$ is sufficiently large.

The Cauchy problem to the system of nonlinear damped wave equations is treated. Several authors have shown existence and asymptotic behavior of global solutions to the above problem when the space dimension is not greater than three. We will show the existence and asymptotic behavior of global solutions to the problem with rapidly decaying initial data when the space dimension is greater than three, where we apply estimates in weighted Sobolev spaces of the above solution operator. Moreover, using the theory of modulation spaces introduced by Feitinger [4], we will also show the existence and asymptotic behavior of global solutions to the problem with slowly decaying initial data.