Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGlobal dynamics above the ground state for the energy-critical Schrodinger equation with radial data
55
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Consider the focusing energy critical Schrodinger equation in three space dimensions with radial initial data in the energy space. We describe the global dynamics of all the solutions of which the energy is at most slightly larger than that of the ground states, according to whether it stays in a neighborhood of them, blows up in finite time or scatters. In analogy with the paper by Schlag and the first author on the subcritical equation, the proof uses an analysis of the hyperbolic dynamics near them and the variational structure far from them. The key step that allows to classify the solutions is the one-pass lemma. The main difference from the subcritical case is that one has to introduce a scaling parameter in order to describe the dynamics near them. One has to take into account this parameter in the analysis around the ground states by introducing some orthogonality conditions. One also has to take it into account in the proof of the one-pass lemma by comparing the contribution in the variational region and in the hyperbolic region.
rate research
Read More
Consider the focusing nonlinear Schrodinger equation with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and solitons with positive large energy, which are unstable. We classify the global dynamics into 9 sets of solutions in the phase space including both solitons, restricted by small mass, radial symmetry, and an energy bound slightly above the second lowest one of solitons. The classification includes a stable set of solutions which start near the first excited solitons, approach the ground states locally in space for large time with large radiation to the spatial infinity, and blow up in negative finite time.
We revisit the problem of scattering below the ground state threshold for the mass-supercritical focusing nonlinear Schrodinger equation in two space dimensions. We present a simple new proof that treats the case of radial initial data. The key ingredient is a localized virial/Morawetz estimate; the radial assumption aids in controlling the error terms resulting from the spatial localization.
We consider the global dynamics below the ground state energy for the Zakharov system in the 3D radial case. We obtain dichotomy between the scattering and the growup.
By definition, the exterior asymptotic energy of a solution to a wave equation on $mathbb{R}^{1+N}$ is the sum of the limits as $tto pminfty$ of the energy in the the exterior ${|x|>|t|}$ of the wave cone. In our previous work (JEMS 2012, arXiv:1003.0625), we have proved that the exterior asymptotic energy of a solution of the linear wave equation in odd space dimension $N$ is bounded from below by the conserved energy of the solution. In this article, we study the analogous problem for the linear wave equation with a potential begin{equation} label{abstractLW} tag{*} partial_t^2u+L_Wu=0,quad L_W:=-Delta -frac{N+2}{N-2}W^{frac{4}{N-2}} end{equation} obtained by linearizing the energy critical wave equation at the ground-state solution $W$, still in odd space dimension. This equation admits nonzero solutions of the form $A+tB$, where $L_WA=L_WB=0$ with vanishing asymptotic exterior energy. We prove that the exterior energy of a solution of eqref{abstractLW} is bounded from below by the energy of the projection of the initial data on the orthogonal complement of the space of initial data corresponding to these solutions. This will be used in a subsequent paper to prove soliton resolution for the energy-critical wave equation with radial data in all odd space dimensions. We also prove analogous results for the linearization of the energy-critical wave equation around a Lorentz transform of $W$, and give applications to the dynamics of the nonlinear equation close to the ground state in space dimensions $3$ and $5$.
Consider a finite energy radial solution to the focusing energy critical semilinear wave equation in 1+4 dimensions. Assume that this solution exhibits type-II behavior, by which we mean that the critical Sobolev norm of the evolution stays bounded on the maximal interval of existence. We prove that along a sequence of times tending to the maximal forward time of existence, the solution decomposes into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space. If, in addition, we assume that the critical norm of the evolution localized to the light cone (the forward light cone in the case of global solutions and the backwards cone in the case of finite time blow-up) is less than 2 times the critical norm of the ground state solution W, then the decomposition holds without a restriction to a subsequence.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
