Related papers: Threshold solutions for the focusing generalized H…
In this work, we consider the focusing generalized inhomogeneous Hartree equation with potential \[ i u_t + \Delta u - V(x)u + \left(I_{\gamma} * |x|^{-b}|u|^{p}\right)|x|^{-b}|u|^{p-2}u = 0, \] where $0<\gamma<3$ and…
We investigate the large-time dynamics of solutions of multi-dimensional reaction-diffusion equations with ignition type nonlinearities. We consider solutions which are in some sense locally persistent at large time and initial data which…
In this paper,we show that spherical bounded energy solution of the defocusing 3D energy critical Schr\"odinger equation with harmonic potential, $(i\partial_t + \frac {\Delta}2+\frac {|x|^2}2)u=|u|^4u$, exits globally and scatters to free…
We investigate the focusing $\dot H^{1/2}$-critical nonlinear Schr\"{o}dinger equation (NLS) of Hartree type $i\partial_t u + \Delta u = -(|\cdot|^{-3} \ast |u|^2)u$ with $\dot H^{1/2}$ radial data in dimension $d = 5$. It is proved that if…
We consider the metrics of the General Relativity, whose energy-momentum tensor has a bounded support where it is continuous except for a finite step across the corresponding boundary surface. As a consequence, the first derivative of the…
We construct a pure two-bubble solution for the focusing, energy-critical Hartree equation in space dimension $N \geq 7$. The constructed solution is spherically symmetric, global in (at least) the negative time direction and asymptotically…
Consider the focusing energy critical Schrodinger equation in three space dimensions with radial initial data in the energy space. We describe the global dynamics of all the solutions of which the energy is at most slightly larger than that…
We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in $d$ dimensions, for focusing and defocusing interactions. We introduce new higher order energy functionals and prove that they are conserved for solutions of energy…
This paper addresses the focusing cubic-quintic nonlinear Schrodinger equation in three space dimensions. Especially, we study the global dynamics of solutions whose energy and mass equal to those of the ground state in the sprits of…
We consider the energy-critical wave maps equation $\mathbb R^{1+2} \to \mathbb S^2$ in the equivariant case, with equivariance degree $k \geq 2$. It is known that initial data of energy $ < 8k\pi$ and topological degree zero leads to…
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…
In this paper, we consider the longtime dynamics of the solutions to focusing energy-critical Schr\"odinger equation with a defocusing energy-subcritical perturbation term under a ground state energy threshold in four spatial dimension.…
We consider the cubic Nonlinear Schroedinger Equation (NLS) in one space dimension, either focusing or defocusing. We prove that the solutions satisfy a-priori local in time Hs bounds in terms of the Hs size of the initial data for s >=-1/4…
Based on the concentration-compactness-rigidity argument in \cite{KenM:NLS,KenM:NLW} and the non-degeneracy of the ground state in \cite{LLTX:Nondeg,LLTX:g-Hart,LTX:Nondeg}, long time dynamics for the focusing energy-critical Hartree…
We examine the energy-critical nonlinear heat equation in critical spaces for any dimension greater or equal than three. The aim of this paper is two-fold. First, we establish a necessary and sufficient condition on initial data at or below…
In this paper, we investigate the optimal large-time behavior of the global solution to a singular chemotaxis system in the whole space $\mathbb{R}^d$ with $d=2,3$. Assuming that the initial data is sufficiently close to an equilibrium…
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…
We study the focusing, cubic, nonlinear Klein-Gordon equation in 3D with large radial data in the energy space. This equation admits a unique positive stationary solution, called the ground state. In 1975, Payne and Sattinger showed that…
We consider energy sub-critical defocusing nonlinear wave equations on $\mathbb{R}^3$ and establish the existence of unique global solutions almost surely with respect to a unit-scale randomization of the initial data on Euclidean space. In…
We prove almost sure global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on $\mathbb{R}^3$ with random initial data in $ H^s(\mathbb{R}^3) \times H^{s-1}(\mathbb{R}^3)$ for $s > \frac 12$. The main new…