Related papers: Introduction to scattering for radial 3D NLKG belo…
We study the 1D Klein-Gordon equation with variable coefficient cubic nonlinearity. This problem exhibits a striking resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the…
We obtain almost-sure scattering for the cubic defocusing Schr{\"o}dinger equation in the Euclidean space {$\mathbb{R}^3$}, with randomized radially-symmetric initial data at some supercritical regularity scales. Since we make no smallness…
We revisit the scattering problems for the 2D mass super-critical Schr\"{o}dinger and Klein-Gordon equations with radial data below the ground state in the energy space. We give an alternative proof of energy scattering for both defocusing…
We prove small energy scattering for the 3D Klein-Gordon-Zakharov system with radial symmetry. The idea of proof is the same as the Zakharov system studied in \cite{GN}, namely to combine the normal form reduction and the radial-improved…
The scattering problem for the Klein-Gordon equation with cubic convolution nonlinearity is considered. Based on the Strichartz estimates for the inhomogeneous Klein-Gordon equation we prove the existence of the scattering operator.
In this paper, we study the focusing and defocusing energy--subcritical, nonlinear wave equation in $\mathbb{R}^{1+d}$ with radial initial data for $d = 4,5$. We prove that if a solution remains bounded in the critical space on its interval…
In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schr\"{o}dinger equation (NLS) with the combined terms iu_t + \Delta u = -|u|^4u + |u|^2u \tag{CNLS} in…
In this paper, we study the global well-posedness and scattering problem in the energy space for both focusing and defocusing the Klein-Gordon-Hartree equation in the spatial dimension $d \geq 3$. The main difficulties are the absence of an…
We consider the question of scattering for the boson star equation in three space dimensions. This is a semi-relativistic Klein-Gordon equation with a cubic nonlinearity of Hartree type. We combine weighted estimates, obtained by exploiting…
We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and…
We study the Cauchy problem for the radial energy critical nonlinear wave equation in three dimensions. Our main result proves almost sure scattering for radial initial data below the energy space. In order to preserve the spherical…
We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in 4 to 6 dimensional spaces with radial initial data. We define $w=r^{(d-1)/2} u$, reduce the equation above to one-dimensional…
We consider the NLS with variable coefficients in dimension $n\ge3$ \begin{equation*} i \partial_t u - Lu +f(u)=0, \qquad Lv=\nabla^{b}\cdot(a(x)\nabla^{b}v)-c(x)v, \qquad \nabla^{b}=\nabla+ib(x), \end{equation*} on $\mathbb{R}^{n}$ or more…
We show that the solutions of the three-dimensional critical defocusing nonlinear wave equation with Neumann boundary conditions outside a ball and radial initial data scatter. This is to our knowledge the first result of scattering for a…
In this paper, we consider the Cauchy problem {align*} \{{array}{ll}&i u_t+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u, \quad t\in\mathbb{R}, \quad x\in\mathbb{R}^N &u(0,x)=\phi(x)\in \Sigma, \quad x\in\mathbb{R}^N, {array}. {align*}…
We solve the Klein-Gordon equation in the presence of a spatially one-dimensional cusp potential. The scattering solutions are obtained in terms of Whittaker functions and the condition for the existence of transmission resonances is…
We consider the scattering results of the radial solutions below the ground state to the focusing inhomogeneous nonlinear Schr\"odinger equation $$i\partial_tu+\Delta u +|x|^{-b}|u|^{p}u=0$$ in two dimension, where $0<b<1$ and…
We consider the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b}|u|^\alpha u = 0\qtq{on}\R\times\R^N, \] with $\alpha=\tfrac{4-2b}{N-2}$, $N=\{3,4,5\}$ and $0<b\leq…
We consider the pure-power defocusing nonlinear Klein-Gordon equation, in the energy subcritical case, posed on the product space $\mathbb R^d\times \mathbb T$, where $\mathbb T$ is the one-dimensional flat torus. In this framework, we…
We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schr\"odinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on…