272 research outputs found
On collapsing ring blow up solutions to the mass supercritical NLS
We consider the nonlinear Schr\"odinger equation i\pa_tu+\Delta
u+u|u|^{p-1}=0 in dimension and in the mass super critical and
energy subcritical range For initial
data with radial symmetry, we prove a universal upper bound on the
blow up speed. We then prove that this bound is sharp and attained on a family
of collapsing ring blow up solutions first formally predicted by Gavish, Fibich
and Wang.Comment: 48 page
Blow up dynamics for smooth equivariant solutions to the energy critical Schr\"odinger map
We consider the energy critical Schr\"odinger map problem with the 2-sphere
target for equivariant initial data of homotopy index . We show the
existence of a codimension one set of smooth well localized initial data
arbitrarily close to the ground state harmonic map in the energy critical norm,
which generates finite time blow up solutions. We give a sharp description of
the corresponding singularity formation which occurs by concentration of a
universal bubble of energy
On the stability of critical chemotactic aggregation
We consider the two dimensional parabolic-elliptic Patlak-Keller-Segel model
of chemotactic aggregation for radially symmetric initial data. We show the
existence of a stable mechanism of singularity formation and obtain a complete
description of the associated aggregation process.Comment: 80 page
LECTURES ON NONLINEAR DISPERSIVE EQUATIONS I
CONTENTS
J. Bona
Derivation and some fundamental properties of nonlinear dispersive waves equations
F. Planchon
Schr\"odinger equations with variable coecients
P. Rapha\"el
On the blow up phenomenon for the L^2 critical non linear Schrodinger Equatio
Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains
We consider a semilinear elliptic problem with a nonlinear term which is the
product of a power and the Riesz potential of a power. This family of equations
includes the Choquard or nonlinear Schroedinger--Newton equation. We show that
for some values of the parameters the equation does not have nontrivial
nonnegative supersolutions in exterior domains. The same techniques yield
optimal decay rates when supersolutions exists.Comment: 47 pages, 8 figure
Study on a class of Schrödinger elliptic system involving a nonlinear operator
This paper considers a class of Schrödinger elliptic system involving a nonlinear operator. Firstly, under the simple condition on and ', we prove the existence of the entire positive bounded radial solutions. Secondly, by using the iterative technique and the method of contradiction, we prove the existence and nonexistence of the entire positive blow-up radial solutions. Our results extend the previous existence and nonexistence results for both the single equation and systems. In the end, we give two examples to illustrate our results
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