Related papers: On the $L^{2}$-critical nonlinear Schrodinger equa…

It is well-known that there are global small data smooth solutions for the 3-D semilinear Klein-Gordon equations $\square u + u = F(u,{\partial u})$ with cubic nonlinearities. However, for the short pulse initial data $(u, \partial_tu)(0,…

We consider the following nonlinear Schr\"{o}dinger equation with an inverse potential: \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u\pm\frac{1}{|x|^{2\sigma}}u=0 \] in $\mathbb{R}^N$. From the classical argument, the…

We study a system of inhomogeneous nonlinear Schr\"odinger equations that emerge in optical media with a $\chi^{(2)}$ nonlinearity. This nonlinearity, whose local strength is subject to a cusp-shaped spatial modulation, $\chi^{(2)}\sim…

For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^1\times L^2$. In particular, any global…

We prove global well-posedness for the L^{2}-critical cubic defocusing nonlinear Schr\"odinger equation on R^{2} with data u_{0} \in H^{s}(R^{2}) for s > {1/3}.

We establish the local existence and the uniqueness of solutions of the heat equation with a nonlinear boundary condition for the initial data in uniformly local $L^r$ spaces. Furthermore, we study the sharp lower estimates of the blow-up…

We consider short-range mass-subcritical nonlinear Schr\"odinger equations and we show that the corresponding solutions with initial data in $\Sigma$ scatter in $H^1$. Hence we up-grade the classical scattering result proved by Yajima and…

In this work, we mainly focus on the energy-supercritical nonlinear Schr\"odinger equation, $$ i\partial_{t}u+\Delta u= \mu|u|^p u, \quad (t,x)\in \mathbb{R}^{d+1}, $$ with $\mu=\pm1$ and $p>\frac4{d-2}$. %In this work, we consider the…

We prove, for the energy critcal, focusing NLW, that for Cauchy data (u_0, u_1) whose energy is smaller than that of (W,0), where W is the well-known radial positive solution to the corresponding ellipyic equation, the following dichotomy…

In this work we consider a system of nonlinear Schr\"odinger equations whose nonlinearities satisfy a power-type-growth. First, we prove that the Cauchy problem is local and global well-posedness in $L^2$ and $H^1$. Next, we establish the…

In this work we study a system of Schr\"odinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in…

We consider the defocusing energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation (INLS) $iu_t + \Delta u = |x|^{-b}|u|^{k}u$ in $\mathbb{R} \times \mathbb{R}^{n}$ where $n \geq 3$, $0<b<\min(2, n/2)$, and $k=(4-2b)/(n-2)$. We…

We prove large-data scattering in $H^1$ for inhomogeneous nonlinear Schr\"odinger equations in two space dimensions for all powers $p>0$. We assume the inhomogeneity is nonnegative and repulsive; we additionally require decay at infinity in…

In this paper, we study the long-time behavior of global solutions to the Schr\"odinger-Choquard equation $$i\partial_tu+\Delta u=-(I_\alpha\ast|\cdot|^b|u|^{p})|\cdot|^b|u|^{p-2}u.$$ Inspired by Murphy, who gave a simple proof of…

In this paper we consider a semilinear parabolic equation with nonlinear and nonlocal boundary condition and nonnegative initial datum. We prove some global existence results. Criteria on this problem which determine whether the solutions…

We establish global well-posedness and scattering for solutions to the defocusing mass-critical (pseudoconformal) nonlinear Schr\"odinger equation $iu_t + \Delta u = |u|^{4/n} u$ for large spherically symmetric $L^2_x(\R^n)$ initial data in…

The initial-boundary value problem for the Schr\"odinger-Korteweg-de Vries system is considered on the left and right half-line for a wide class of initial-boundary data, including the energy regularity $H^1(\R^{\pm})\times H^1(\R^{\pm})$…

In this paper we consider the inhomogeneous nonlinear Schr\"odinger equation $i\partial_t u +\Delta u=K(x)|u|^\alpha u,\, u(0)=u_0\in H^s({\mathbb R}^N),\, s=0,\,1,$ $N\geq 1,$ $|K(x)|+|x|^s|\nabla^sK(x)|\lesssim |x|^{-b},$…

In the first part of this paper, we investigate the sharp threshold of blow-up and global existence for the focusing nonlinear Schr\"{o}dinger equation with combined nonlinearities of mass-critical and mass-subcritical power-type.…

We consider the inhomogeneous nonlinear Schr\"odinger equation with inverse-square potential in $\mathbb{R}^N$ $$ i u_t + \mathcal{L}_a u+\lambda |x|^{-b}|u|^\alpha u = 0,\;\;\mathcal{L}_a=\Delta -\frac{a}{|x|^2}, $$ where $\lambda=\pm1$,…