Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS

@article{Raphael2010ExistenceAU,
  title={Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS},
  author={Pierre Raphael and J{\'e}r{\'e}mie Szeftel},
  journal={Journal of the American Mathematical Society},
  year={2010},
  volume={24},
  pages={471-546}
}
We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial_tu+\Delta u+k(x)|u|^{2}u=0$. From standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2} M_k$. In this paper, we consider the dynamics at threshold $\|u_0\|_{L^2}=M_k$ and give a necessary and sufficient condition on $k$ to ensure the existence of critical mass finite time blow up elements. Moreover, we give a complete classification in the energy… 

The instability of Bourgain-Wang solutions for the L2 critical NLS

We consider the two dimensional $L^2$ critical nonlinear Schr\"odinger equation $i\partial_tu+\Delta u+uu^2=0$. In their pioneering 1997 work, Bourgain and Wang have constructed smooth solutions

On asymptotic dynamics for L2 critical generalized KdV equations with a saturated perturbation

In this paper, we consider the $L^2$ critical gKdV equation with a saturated perturbation: $\partial_t u+(u_{xx}+u^5-\gamma u|u|^{q-1})_x=0$, where $q>5$ and $0<\gamma\ll1$. For any initial data

Classification of minimal mass blow-up solutions for an $${L^{2}}$$L2 critical inhomogeneous NLS

AbstractWe establish the classification of minimal mass blow-up solutions of the $${L^{2}}$$L2 critical inhomogeneous nonlinear Schrödinger equation $$i\partial_t u + \Delta u +

Minimal mass blow-up solutions for the $L^2$-critical NLS with the Delta potential for radial data in one dimension

Abstract. We consider the L-critical nonlinear Schrödinger equation (NLS) with the delta potential i∂tu+ ∂ 2 xu+ μδu + |u| u = 0, t ∈ R, x ∈ R, where μ ∈ R, and δ is the Dirac delta distribution at x

Nondispersive solutions to the L2-critical Half-Wave Equation

AbstractWe consider the focusing L2-critical half-wave equation in one space dimension, $$i \partial_t u = D u - |u|^2 u$$, where D denotes the first-order fractional derivative. Standard arguments

Type II blow up for the energy supercritical NLS

We consider the energy super critical nonlinear Schr\"odinger equation $$i\pa_tu+\Delta u+u|u|^{p-1}=0$$ in large dimensions $d\geq 11$ with spherically symmetric data. For all $p>p(d)$ large enough,

Minimal mass blow-up solutions for nonlinear Schr\"{o}dinger equations with a Hartree nonlinearity

We consider the following nonlinear Schr\"{o}dinger equation with a Hartree nonlinearity: \[ i\frac{\partial u}{\partial t}+\Delta

Remarks on minimal mass blow up solutions for a double power nonlinear Schr\"{o}dinger equation

We consider the following nonlinear Schrodinger equation with double power nonlinearity \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u+|u|^{p-1}u=0,\quad 1<p<1+\frac{4}{N} \] in

Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation

AbstractWe consider the supercritical inhomogeneous nonlinear Schrödinger equation $$i\partial_t u+\Delta u+|x|^{-b}|u|^{2\sigma}u=0,$$i∂tu+Δu+|x|-b|u|2σu=0,where $${(2 - b)/N < \sigma < (2 -
...

References

SHOWING 1-10 OF 34 REFERENCES

On Stability of Pseudo-Conformal Blowup for L2-critical Hartree NLS

AbstractWe consider L2-critical focusing nonlinear Schrödinger equations with Hartree type nonlinearity $$i \partial_{t} u = - \Delta u - \left(\Phi \ast |u|^2 \right) u \quad {\rm in}\, \mathbb

Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation

AbstractWe consider the critical nonlinear Schrödinger equation $iu_{t} = -\Delta u-|u|^{4/N}$ with initial condition u(0, x) = u0.For u0$\in$H1, local existence in time of solutions on an interval

Characterization of Minimal-Mass Blowup Solutions to the Focusing Mass-Critical NLS

It is proved that in dimensions $d\geq4$, the only spherically symmetric minimal-mass nonscattering solutions are, up to phase rotation and scaling, the pseudoconformal ground state and the ground state solitary wave.

Existence and Stability of the log–log Blow-up Dynamics for the L2-Critical Nonlinear Schrödinger Equation in a Domain

Abstract.Let $$iu_{t} = -\Delta u-|u|^{\frac{4}{N}}u$$ be the L2-critical nonlinear Schrödinger equation, in a domain $$\Omega \subset \mathbb{R}^{N}$$ with initial data in $$H^{1}_{0}(\Omega)$$

Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation

The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term.

ON A SHARP LOWER BOUND ON THE BLOW-UP RATE FOR THE L CRITICAL NONLINEAR SCHRÖDINGER EQUATION

(1.1) (NLS) { iut = −∆u − |u| 4 N u, (t, x) ∈ [0, T ) × R u(0, x) = u0(x), u0 : R → C with u0 ∈ H = H(R ) in dimension N ≥ 1. From a result of Ginibre and Velo [7], (1.1) is locally well-posed in H

Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity

AbstractWe consider the nonlinear Schrödinger equation:(1) $${{i\partial u} \mathord{\left/ {\vphantom {{i\partial u} {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}} = - \Delta u -

On universality of blow-up profile for L2 critical nonlinear Schrödinger equation

We consider finite time blow-up solutions to the critical nonlinear Schrödinger equation iut=-Δu-|u|4/Nu with initial condition u0∈H1. Existence of such solutions is known, but the complete blow-up

On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions d⩾2

AbstractFor the focusing mass-critical NLS $$iu_t + \Delta u = - \left| u \right|^{\tfrac{4} {d}} u$$, it is conjectured that the only global non-scattering solution with ground state mass must be a