相关论文: Global existence and scattering for rough solution…

We consider the Cauchy problem of the cubic nonlinear Schr\"odinger equation (NLS) on $\mathbb R^d$, $d \geq 3$, with random initial data and prove almost sure well-posedness results below the scaling critical regularity $s_\text{crit} =…

In this paper, we investigate the global well-posedness and scattering theory for the defocusing nonlinear Schr\"odinger equation $iu_t + \Delta_\Omega u = |u|^\alpha u$ in the exterior domain $\Omega$ of a smooth, compact and strictly…

In this article, we prove the global well-posedness and scattering of the cubic focusing infinite coupled nonlinear Schr\"odinger system on $\mathbb{R}^2$ below the threshold in $L_x^2h^1(\mathbb{R}^2\times \mathbb{Z})$. We first establish…

This paper is concerned with the Cauchy problem for an inhomogeneous nonlinear Schrodinger equation with exponential growth nonlinearity and harmonic potential in two space dimensions. We prove global well-posedness, existence of the…

The initial value problem for some defocusing coupled nonlinear Schrodinger equations is investigated. Global well-posedness and scattering are established.

We study the energy-critical nonlinear Schr\"{o}dinger equation with randomised initial data in dimensions $d>6$. We prove that the Cauchy problem is almost surely globally well-posed with scattering for randomised super-critical initial…

We study the wellposedness of Cauchy problem for the fourth order nonlinear Schr\"odinger equations i\partial_t u=-\eps\Delta u+\Delta^2 u+P((\partial_x^\alpha u)_{\abs{\alpha}\ls 2}, (\partial_x^\alpha \bar{u})_{\abs{\alpha}\ls 2}),\quad…

In this paper, we consider the defocusing mass-critical nonlinear fourth-order Schr\"odinger equation. Using the $I$-method combined with the interaction Morawetz estimate, we prove that the problem is globally well-posed in…

We prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schr\"odinger equation. Precisely we show that a unique and global solution exists for initial data in the Sobolev space…

We consider the Cauchy problem for the defocusing nonlinear Schr\"odinger equations (NLS) on the real line with a special subclass of almost periodic functions as initial data. In particular, we prove global existence of solutions to NLS…

In this paper we prove that the focusing, $d$-dimensional mass critical nonlinear Schr{\"o}dinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R}^{d})$, $\| u_{0} \|_{L^{2}(\mathbf{R}^{d})} < \| Q…

In this paper, we consider the defocusing nonlinear Schr\"odinger equation in space dimensions $d\geq 4$. We prove that if $u$ is a radial solution which is \emph{priori} bounded in the critical Sobolev space, that is, $u\in L_t^\infty…

In this paper, we consider the fourth-order Schr\"odinger equations with focusing, $L^2$-supercritical nonlinearity in one dimension. We prove the global existence and scattering of solutions below the ground state threshold under the…

We consider the defocusing energy-critical nonlinear Schr\"odinger equation of fourth order $iu_t+\Delta^2 u=-|u|^\frac{8}{d-4}u$. We prove that any finite energy solution is global and scatters both forward and backward in time in…

In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term $\nu (-\Delta)^2 u_t$, where $\nu >0$ is a constant. As being mentioned in [8,10], the linear principal part brings both the diffusion…

We consider the dispersion managed nonlinear Schr\"dinger equations with quintic and cubic nonlinearities in one and two dimensions, respectively. We prove the global well-posedness and scattering in $L_x^2$ for small initial data employing…

We consider a 3d cubic focusing nonlinear Schr\"odinger equation with a potential $$i\partial_t u+\Delta u-Vu+|u|^2u=0,$$ where $V$ is a real-valued short-range potential having a small negative part. We find criteria for global…

In this paper, we study the Cauchy problem for the nonlinear Schr\"odinger equations with Coulomb potential $i\partial_tu+\Delta u+\frac{K}{|x|}u=\lambda|u|^{p-1}u$ with $1<p\leq5$ on $\mathbb{R}^3$. We mainly consider the influence of the…

We study the Cauhcy problem for space-time fractional nonlinear Schr\"odinger equation with a general nonlinearity. We prove the local well-posedness of it in fractional Sobolev spaces based on the decay estimates and H\"older type…

We consider the Cauchy problem for linearly damped nonlinear Schr\"odinger equations \[ i\partial_t u + \Delta u + i a u= \pm |u|^\alpha u, \quad (t,x) \in [0,\infty) \times \mathbb{R}^N, \] where $a>0$ and $\alpha>0$. We prove the global…