Journal ArticleDOI
Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schrödinger Equation
TLDR
In this paper, the authors consider finite time blow up solutions to the critical nonlinear Schrodinger equation, and prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of L2 mass at the blow up point, the second part correspond to the regular part and has a strong L2 limit at blow up time.Abstract:
We consider finite time blow up solutions to the critical nonlinear Schrodinger equation Open image in new window For a suitable class of initial data in the energy space H1, we prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of L2 mass at the blow up point, the second part corresponds to the regular part and has a strong L2 limit at blow up time.read more
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Blow-up for the 1D nonlinear Schr\"odinger equation with point nonlinearity I: Basic theory
Justin Holmer,Chang Liu +1 more
TL;DR: In this article, the 1D nonlinear Schrodinger equation (NLS) with focusing point nonlinearity was considered and a sharp Gagliardo-Nirenberg inequality and a local virial identity were derived.
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Periodic solutions of nonlinear Schrödinger equations: a paradifferential approach
Journal ArticleDOI
Blow-up solutions on a sphere for the 3D quintic NLS in the energy space
Justin Holmer,Svetlana Roudenko +1 more
TL;DR: In this article, it was shown that if u(t) is a log-log blow-up solution, of the type studied by Merle and Raphael, to the L2 critical focusing NLS equation i∂tu+Δu+|u|4∕du=0 with initial data u0∈H1(ℝd) in the cases d=1,2, then u t remains bounded in H1 away from the blowup point.
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Variational approach to complicated similarity solutions pf higher-order nonlinear PDEs. I
TL;DR: In this paper, higher-order nonlinear parabolic hyperbolic and nonlinear dispersion equations admit exact blow-up or compacton solutions, which are induced by elliptic equations with non-Lipschitz nonlinearities.
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Lectures on the Energy Critical Nonlinear Wave Equation
TL;DR: The road map of the local theory of the Cauchy problem can be found in this paper, where the concentration compactness/rigidity theorem method for critical problems is presented.
References
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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.
Journal ArticleDOI
Nonlinear Schrödinger equations and sharp interpolation estimates
TL;DR: In this paper, a sharp sufficient condition for global existence for the nonlinear Schrodinger equation is obtained for the case σ = 2/N. This condition is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of Nirenberg and Gagliardo.
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