Journal ArticleDOI
Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation
Frank Merle,Pierre Raphaël +1 more
TLDR
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.Abstract:
We consider the critical nonlinear Schrodinger equation $iu_{t} = -\Delta u-|u|^{4/N}$ with initial condition u(0, x) = u0read more
Citations
More filters
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
The blow-up dynamic and upper bound on the blow-up rate for 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 in dimension N = 1 is considered, and local existence in the time of solutions on an interval [0, T] is known, and there exist finite time blowup solutions, that is, u0 such that limt.T 1.
Journal ArticleDOI
On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation
TL;DR: 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.
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.
References
More filters
Book
Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
Book ChapterDOI
Elliptic Partial Differential Equations of Second Order
Piero Bassanini,Alan R. Elcrat +1 more
TL;DR: In this paper, a class of partial differential equations that generalize and are represented by Laplace's equation was studied. And the authors used the notation D i u, D ij u for partial derivatives with respect to x i and x i, x j and the summation convention on repeated indices.
Journal Article
Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media
Journal ArticleDOI
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
TL;DR: In this article, a constrained minimization method was proposed for the case of dimension N = 1 (Necessary and sufficient conditions) for the zero mass case, where N is the number of dimensions in the dimension N.
Related Papers (5)
On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation
The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation
Frank Merle,Pierre Raphaël +1 more