167 research outputs found
Blow-up of a critical Sobolev norm for energy-subcritical and energy-supercritical wave equations
This work concerns the semilinear wave equation in three space dimensions
with a power-like nonlinearity which is greater than cubic, and not quintic
(i.e. not energy-critical). We prove that a scale-invariant Sobolev norm of any
non-scattering solution goes to infinity at the maximal time of existence. This
gives a refinement on known results on energy-subcritical and
energy-supercritical wave equation, with a unified proof.
The proof relies on the channel of energy method, as in arXiv:1204.0031, in
weighted scale-invariant Sobolev spaces which were introduced in
arXiv:1506.00788. These spaces are local, thus adapted to finite speed of
propagation, and related to a conservation law of the linear wave equation. We
also construct the adapted profile decomposition
Weighted Strichartz estimates for radial Schr\"odinger equation on noncompact manifolds
We prove global weighted Strichartz estimates for radial solutions of linear
Schr\"odinger equation on a class of rotationally symmetric noncompact
manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces.
This yields classical Strichartz estimates with a larger class of exponents
than in the Euclidian case and improvements for the scattering theory. The
manifolds, whose volume element grows polynomially or exponentially at
infinity, are characterized essentially by negativity conditions on the
curvature, which shows in particular that the rich algebraic structure of the
Hyperbolic and Damek-Ricci spaces is not the cause of the improved dispersive
properties of the equation. The proofs are based on known dispersive results
for the equation with potential on the Euclidean space, and on a new one, valid
for C^1 potentials decaying like 1/r^2 at infinity
Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation
We study a damped semi-linear wave equation in a bounded domain with smooth
boundary. It is proved that any sufficiently smooth solution can be stabilised
locally by a finite-dimensional feedback control supported by a given open
subset satisfying a geometric condition. The proof is based on an investigation
of the linearised equation, for which we construct a stabilising control
satisfying the required properties. We next prove that the same control
stabilises locally the non-linear problem.Comment: 29 page
Maximizers for the Strichartz norm for small solutions of mass-critical NLS
Consider the mass-critical nonlinear Schr\"odinger equations in both focusing
and defocusing cases for initial data in in space dimension N. By
Strichartz inequality, solutions to the corresponding linear problem belong to
a global space in the time and space variables, where . In 1D
and 2D, the best constant for the Strichartz inequality was computed by
D.~Foschi who has also shown that the maximizers are the solutions with
Gaussian initial data.
Solutions to the nonlinear problem with small initial data in are
globally defined and belong to the same global space. In this work we
show that the maximum of the norm is attained for a given small mass. In
addition, in 1D and 2D, we show that the maximizer is unique and obtain a
precise estimate of the maximum. In order to prove this we show that the
maximum for the linear problem in 1D and 2D is nondegenerated.Comment: To be published in Annali della Scuola Normale Superiore di Pis
- …