50 research outputs found
Maximum Decay Rate for the Nonlinear Schr\"odinger Equation
In this paper, we consider global solutions for the following nonlinear
Schr\"odinger equation in with
and if
We show that no nontrivial solution can decay faster than the solutions of the
free Schr\"odinger equation, provided that lies in the weighted Sobolev
space in the energy space, namely
or in according to the different cases
A generalized interpolation inequality and its application to the stabilization of damped equations.
Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation
International audienceIn this paper, we consider the nonlinear Schrödinger equation with the super critical power of nonlinearity in the attractive case. We give a sufficient condition and a necessary condition to obtain global or blowing up solutions. These conditions coincide in the critical case, thereby extending the results of Weinstein \cite{MR84d:35140,MR87i:35026}. Furthermore, we improve a blow-up condition
The dual space of a complex Banach space restricted to the field of real numbers
Solutions of some partial differential equations are obtained as critical
points of a real funtional. Then the Banach space where this functional is
defined has to be real, otherwise, it is not differentiable. It follows that
the equation is solved with respect to the real dual space of this Banach
space. But if the solution is complex-valued there is the following problem:
what does the multiplication of this equation by a complex number mean ? In
this note, we explain how to rigorously define this operation