On the existence of maximizers for a family of Restriction Theorems

We prove the existence of maximizers for a general family of restrictions operators, up to the end-point. We also provide some counterxamples in the end-point case.Comment: 6 page

Semiclassical resolvent estimates at trapped sets

We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs microlocally supported away from the trapping, a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs to be supported at the trapped set, giving estimates which are intermediate between the trapping and non-trapping ones.Comment: 5 page

Existence of maximizers for Sobolev-Strichartz inequalities

We prove the existence of maximizers of Sobolev-Strichartz estimates for a general class of propagators, involving relevant examples, as for instance the wave, Dirac and the hyperbolic Schrodinger flows.Comment: 10 page

Global attractors for the one dimensional wave equation with displacement dependent damping

We study the long-time behavior of solutions of the one dimensional wave equation with nonlinear damping coefficient. We prove that if the damping coefficient function is strictly positive near the origin then this equation possesses a global attractor