Related papers: Standing Ring Blowup Solutions for Cubic NLS

We study the focusing $L^2$-critical and supercritical stochastic nonlinear Schr\"odinger equation subject to additive or multiplicative noise. We investigate global or long time behavior of solutions in $H^1$, which would correspond to…

We consider in this paper blow-up solutions of the semilinear wave equation in one space dimension, with an exponential source term. Assuming that initial data are in $H^{1}_{loc}\times L^2_{loc}$ or some times in $ W^{1,\infty}\times…

We investigate the focusing and defocusing energy-critical stochastic nonlinear Schr\"odinger equation, subject to random perturbations in the form of either additive or multiplicative (Stratonovich) noise. We establish local well-posedness…

We consider the defocusing $\dot{H}^{1/2}$-critical nonlinear Schr\"odinger equation in dimensions $d\geq 5$. In the spirit of Kenig and Merle [Trans. Amer. Math. Soc. 362 (2010), 1937--1962], we combine a concentration-compactness approach…

For superlinear heat equations with the Dirichlet boundary condition, the $L^\infty$ estimates of radially symmetric solutions are studied. In particular, the uniform boundedness of global solutions and the non-existence of solutions with…

We consider the two dimensional $L^2$ critical nonlinear Schr\"odinger equation $i\pa_tu+\Delta u+u|u|^2=0$. In the pioneering work \cite{BW}, Bourgain and Wang have constructed smooth solutions which blow up in finite time $T<+\infty$ with…

We consider finite time blowup solutions of the $L^2$-critical cubic focusing nonlinear Schr\"odinger equation on $\R^2$. Such functions, when in $H^1$, are known to concentrate a fixed $L^2$-mass (the mass of the ground state) at the point…

We consider NLS on $\T^2$ with multiplicative spatial white noise and nonlinearity between cubic and quartic. We prove global existence, uniqueness and convergence almost surely of solutions to a family of properly regularized and…

We consider the mass concentration phenomenon for the $L^2$-critical nonlinear Schr\"odinger equations of higher orders. We show that any solution $u$ to $iu_{t} + (-\Delta)^{\frac\alpha 2} u =\pm |u|^\frac{2\alpha}{d}u$, $u(0,\cdot)\in…

We prove polynomial bounds on the growth of higher Sobolev norms for the nonlinear Schrodinger equation set on a torus, in dimension 3, with super-cubic and sub-quintic nonlinearity. Due to improved Strichartz estimates, these bounds are…

In this paper we study the local and global regularity properties of the cubic nonlinear Schr\"odinger equation (NLS) on the half line with rough initial data. These properties include local and global wellposedness results, local and…

We consider the supercritical inhomogeneous nonlinear Schr\"odinger equation (INLS) $$i\partial_t u+\Delta u+|x|^{-b}|u|^{2\sigma}u=0,$$ where $(2-b)/N<\sigma<(2-b)/(N-2)$ and $0<b<\min\{2,N\}$. We prove a Gagliardo-Nirenberg type estimate…

In this paper, we consider the mass-critical nonlinear Schr\"odinger equation in one dimension. Ogawa--Tsutsumi [19] proved a blow-up result for negative energy solution by using a scaling argument for initial data. By the reason, the…

We study the focusing nonlinear Schr\"odinger equation in the $L^2$-supercritical regime with finite energy and finite variance initial data. We investigate solutions above the energy (or mass-energy) threshold. In our first result, we…

We consider the inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{R}^N$ \begin{align}\label{inls} i \partial_t u +\Delta u +V(x)|u|^{\frac{4-2b}{N}}u = 0, \end{align} where $V(x) = k(x)|x|^{-b}$, with $b>0$. Under suitable…

We present a numerical study of solutions to the $2d$ cubic and quintic focusing nonlinear Schr\"odinger equation in the exterior of a smooth, compact and strictly convex obstacle (a disk) with Dirichlet boundary condition. We first…

We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + \lambda c(u)c'(u)( u_x)^2$ with the real parameter $\lambda$. In previous works, it was reported that there exist…

We consider fractional NLS with focusing power-type nonlinearity $$i \partial_t u = (-\Delta)^s u - |u|^{2 \sigma} u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^N,$$ where $1/2< s < 1$ and $0 < \sigma < \infty$ for $s \geq N/2$ and $0 <…

This article is concerned with the small data problem for the cubic nonlinear Schr\"odinger equation (NLS) in one space dimension, and short range modifications of it. We provide a new, simpler approach in order to prove that global…

In this paper, the well-posedness of Cauchy's problem of fractional Schr\"odinger equations with a power type nonlinearity on $n$-dimensional manifolds with nonnegative Ricci curvature is studied. Under suitable volume conditions, the local…