Small data scattering for the nonlinear Schr\"odinger equation on product spaces

We consider the cubic nonlinear Schr\"odinger equation, posed on $\R^n\times M$, where $M$ is a compact Riemannian manifold and $n\geq 2$. We prove that under a suitable smallness in Sobolev spaces condition on the data there exists a unique global solution which scatters to a free solution for large times.Comment: 10 pages, slightly revised version, to appear on Comm. PD

Max-Min characterization of the mountain pass energy level for a class of variational problems

We provide a max-min characterization of the mountain pass energy level for a family of variational problems. As a consequence we deduce the mountain pass structure of solutions to suitable PDEs, whose existence follows from classical minimization argument

On the Local Smoothing for the Schroedinger Equation

We prove a family of identities that involve the solutions to the free Schreodinger equation. As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate and a uniqueness criterion

On the Local Smoothing for a class of conformally invariant Schroedinger equations

We present some a - priori bounds from above and from below for solutions to a class of conformally invariant Schroedinger equations. As a by - product we deduce some new uniqueness results

An Improvement on the Br\'ezis-Gallou\"et technique for 2D NLS and 1D half-wave equation

We revise the classical approach by Br\'ezis-Gallou\"et to prove global well posedness for nonlinear evolution equations. In particular we prove global well--posedness for the quartic NLS posed on general domains $\Omega$ in $\R^2$ with initial data in $H^2(\Omega)\cap H^1_0(\Omega)$, and for the quartic nonlinear half-wave equation on $\R$ with initial data in $H^1(\R)$

Dispersive Estimate for the Wave Equation with Short-Range Potential

In this paper we consider a potential type perturbation of the three dimensional wave equation: $\Box u + V(x)u = 0 u(x, 0) = 0, \partial_t u(x, 0) = f$, where the potential $V \geq 0$ satisfies the following decay assumption: $|V (x)| \leq \frac{C}{1+|x|^{2+\epsilon_0}}$, for some C, $\epsilon_0 > 0$. We establish some dispersive estimates for the associated propagator

On the decay of solutions to a class of defocusing NLS

We consider the following family of Cauchy problems: {equation*} i\partial_t u= \Delta u - u|u|^\alpha, (t,x) \in \R \times \R^d {equation*} $$u(0)=\varphi\in H^1(\R^d)$$ where $0<\alpha<\frac 4{d-2}$ for $d\geq 3$ and $0<\alpha<\infty$ for $d=1,2$. We prove that the $L^r$-norms of the solutions decay as $t\to \pm \infty$, provided that $2<r<\frac{2d}{d-2}$ when $d\geq 3$ and $2<r<\infty$ when $d=1,2$. In particular we extend previous results obtained by Ginibre and Velo for $d\geq 3$ and by Nakanishi for $d=1,2$, where the same decay results are proved under the extra assumption $\alpha >\frac 4d$.Comment: