Journal ArticleDOI
On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation
TLDR
In this paper, the authors considered finite time blow-up solutions to the critical nonlinear Schrodinger equation iut=-Δu-|u|4/Nu with initial condition u0∈H1.Abstract:
We consider finite time blow-up solutions to the critical nonlinear Schrodinger equation iut=-Δu-|u|4/Nu with initial condition u0∈H1. Existence of such solutions is known, but the complete blow-up dynamic is not understood so far. For a specific set of initial data, finite time blow-up with a universal sharp upper bound on the blow-up rate has been proved in [22], [23].read more
Citations
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MonographDOI
Nonlinear dispersive equations : local and global analysis
TL;DR: In this paper, the Korteweg de Vries equation was used for ground state construction in the context of semilinear dispersive equations and wave maps from harmonic analysis.
Journal ArticleDOI
Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation
Frank Merle,Pierre Raphaël +1 more
TL;DR: In this article, the critical nonlinear Schrodinger equation with initial condition u(0, x) = u0 was considered and the initial condition was obtained for the case where x = 0.
Journal ArticleDOI
Renormalization and blow up for charge one equivariant critical wave maps
TL;DR: In this paper, the authors prove the existence of equivariant finite-time blow-up solutions for the wave map problem from ℝ2+1→S petertodd 2 of the form $u(t,r)=Q(\lambda(t)r)+\mathcal{R}( t,r)$cffff where u is the polar angle on the sphere, $Q(r)=2\arctan r$cffff is the ground state harmonic map, λ(t)=t -1-ν, and $\mathcal {R} (t
Journal ArticleDOI
The cubic nonlinear Schrödinger equation in two dimensions with radial data
TL;DR: In this article, the authors establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrodinger equation iut + 1u = ±|u|2u for large spherically symmetric Lx(R) initial data; in the focusing case, of course, that the mass is strictly less than that of the ground state.
Global solutions of nonlinear schrodinger equations
TL;DR: In this paper, the existence of global solutions to the Cauchy problem of nonlinear SchrSdinger equation by establishing time weight function spaces and using the contraction mapping principle was studied.
References
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Symmetry and related properties via the maximum principle
TL;DR: In this paper, the authors show that positive solutions of second order elliptic equations are symmetric about the limiting plane, and that the solution is symmetric in bounded domains and in the entire space.
Journal ArticleDOI
Nonlinear scalar field equations, I existence of a ground state
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The nonlinear Schrödinger equation : self-focusing and wave collapse
TL;DR: In this article, the authors present a basic framework to understand structural properties and long-time behavior of standing wave solutions and their relationship to a mean field generation and acoustic wave coupling.
Journal ArticleDOI
On a class of nonlinear Schro¨dinger equations
TL;DR: In this paper, the existence of standing wave solutions of nonlinear Schrodinger equations was studied and sufficient conditions for nontrivial solutionsu ∈W¯¯¯¯1,2(ℝ�姫 n ) were established.
Journal ArticleDOI
Uniqueness of positive solutions of Δu−u+up=0 in Rn
TL;DR: In this article, the uniqueness of the positive, radially symmetric solution to the differential equation Δu−u+up=0 (with p>1) in a bounded or unbounded annular region in Rn for all n ≥ 1, with the Neumann boundary condition on the inner ball and the Dirichlet boundary condition decaying to zero in the case of an unbounded region, was established.
Related Papers (5)
The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation
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