Global wellposedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case
نویسندگان
چکیده
We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order iut + ∆ u = |u| 8 d−4u. We prove that if a maximal-lifespan radial solution u : I × R → C obeys sup t∈I ‖∆u(t)‖2 < ‖∆W‖2, then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state W at some point in time, then the solution is global and scatters.
منابع مشابه
The Mass-critical Fourth-order Schrödinger Equation in High Dimensions
We prove global wellposedness and scattering for the Mass-critical homogeneous fourth-order Schrödinger equation in high dimensions n ≥ 5, for general L initial data in the defocusing case, and for general initial data with Mass less than certain fraction of the Mass of the Ground State in the focusing case.
متن کاملGlobal Wellposedness of Defocusing Critical Nonlinear Schrödinger Equation in the Radial Case
is globally wellposed in time. More precisely, we obtain a unique solution u = uφ ∈ CH1([0,∞[) such that for all time, u(t) depends continuously on the data φ (in fact, the dependence is even real analytic here). Moreover, there is scattering for t→∞. The same statement holds for radial data φ ∈ H, s ≥ 1 and proves in particular global existence of classical solutions in the radially symmetric ...
متن کاملGlobal well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data
Abstract: For n > 3, we study the Cauchy problem for the fourth order nonlinear Schrödinger equations, for which the existence of the scattering operators and the global well-posedness of solutions with small data in Besov spaces Bs 2,1(R n) are obtained. In one spatial dimension, we get the global well-posedness result with small data in the critical homogeneous Besov spaces Ḃs 2,1. As a by-pr...
متن کاملNonlinear Schrödinger Equation on Real Hyperbolic Spaces
We consider the Schrödinger equation with no radial assumption on real hyperbolic spaces. We obtain sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we obtain strong wellposedness results for NLS. Specifically, for small intial data, we prove L 2 and H 1 global wellposedness for any subcritical nonlinearity (in contrast with the flat case...
متن کاملGlobal Well-posedness and Scattering for the Focusing Nonlinear Schrödinger Equation in the Nonradial Case
The energy-critical, focusing nonlinear Schrödinger equation in the nonradial case reads as follows: i∂tu = −∆u− |u| 4 N−2 u, u(x, 0) = u0 ∈ H(R ), N ≥ 3. Under a suitable assumption on the maximal strong solution, using a compactness argument and a virial identity, we establish the global well-posedness and scattering in the nonradial case, which gives a positive answer to one open problem pro...
متن کامل