102 research outputs found
Small data scattering for the nonlinear Schr\"odinger equation on product spaces
We consider the cubic nonlinear Schr\"odinger equation, posed on , where is a compact Riemannian manifold and . 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 in
with initial data in , and for the quartic
nonlinear half-wave equation on with initial data in
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: , where the potential satisfies the following decay assumption: , for some C, . 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*}
where for and
for . We prove that the -norms of the solutions
decay as , provided that when
and when . In particular we extend previous results
obtained by Ginibre and Velo for and by Nakanishi for , where
the same decay results are proved under the extra assumption .Comment:
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