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The initial value problem for the $L^{2}$ critical semilinear Schrodinger equation in $R^n, n geq 3$ is considered. We show that the problem is globally well posed in $H^{s}({Bbb R^{n}})$ when $1>s>frac{sqrt{7}-1}{3}$ for $n=3$, and when $1>s> frac{-(n-2)+sqrt{(n-2)^2+8(n-2)}}{4}$ for $n geq 4$. We use the ``$I$-method combined with a local in time Morawetz estimate.
We consider the derivative nonlinear Schrodinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and $L^2$-critical with respect to scaling. The first question we discuss is
The initial-boundary value problem (IBVP) for the nonlinear Schrodinger (NLS) equation on the half-plane with nonzero boundary data is studied by advancing a novel approach recently developed for the well-posedness of the cubic NLS on the half-line,
We study the Cauchy problem for the semilinear heat equation with the singular potential, called the Hardy-Sobolev parabolic equation, in the energy space. The aim of this paper is to determine a necessary and sufficient condition on initial data bel
In this paper, we investigate the one-dimensional derivative nonlinear Schrodinger equations of the form $iu_t-u_{xx}+ilambdaabs{u}^k u_x=0$ with non-zero $lambdain Real$ and any real number $kgs 5$. We establish the local well-posedness of the Cauch
We prove global well-posedness for 3D Dirac equation with a concentrated nonlinearity.