相关论文: Global dynamics away from the ground state for the…

We construct a two-parameter continuum of type II blow up solutions for the energy-critical focusing NLS in dimension $ d = 3$. The solutions collapse to a single energy bubble in finite time, precisely they have the form $ u(t,x) = e^{i…

In this article we discuss the long-time dynamics of the radial solutions to the energy-critical wave equation in 3-dimensional space. Given a solution defined for all time $t\geq 0$, we show that the soliton resolution phenomenon happens…

We consider the focusing energy-critical nonlinear wave equation for radially symmetric initial data in space dimensions $D \ge 4$. This equation has a unique (up to sign and scale) nontrivial, finite energy stationary solution $W$, called…

Consider the focusing energy-critical wave equation in space dimension 3, 4 or 5. We prove that any global solution which is bounded in the energy space converges in the exterior of wave cones to a radiation term which is a solution of the…

Consider the focusing energy-critical Klein-Gordon equation in dimension d=3,4,5. We describe the global dynamics of real-valued solutions of which the energy is slightly larger than that of the ground states'. We classify the flows of the…

We investigate the following inhomogeneous nonlinear Schr\"odinger equation in the radial regime, featuring a focusing energy-critical nonlinearity and a defocusing perturbation: $$ i\partial_t u +\Delta u =|x|^{-a} |u|^{p-2} u - |x|^{-b}…

We consider the global dynamics below the ground state energy for the Klein-Gordon-Zakharov system in the 3D radial case; and obtain the dichotomy between scattering and finite time blow up.

We consider the nonlinear Schr\"{o}dinger equation with $L^{2}$-supercritical and $H^{1}$-subcritical power type nonlinearity. Duyckaerts and Roudenko and Campos, Farah, and Roudenko studied the global dynamics of the solutions with same…

In this work we consider a wide range of energy critical wave equation in 3-dimensional space with radial data. We are interested in exterior scattering phenomenon, in which the asymptotic behaviour of a solutions $u$ to the non-linear wave…

We present some numerical findings concerning the nature of the blowup vs. global existence dichotomy for the focusing cubic nonlinear Klein-Gordon equation in three dimensions for radial data. The context of this study is provided by the…

We consider the semilinear wave equation in the radial case with conformal subcritical power nonlinearity. If we consider a blow-up point different from the origin, then we exhibit a new Lyapunov functional which is a perturbation of the…

This paper is concerned with a cubic nonlinear Schr\"odinger system modeling the interaction between an optical beam and its third harmonic in a material with Kerr-type nonlinear response. We are mainly interested in the so-called…

In this article we study global existence and blow-up of solutions for a general class of nonlocal nonlinear wave equations with power-type nonlinearities, $u_{tt}-Lu_{xx}=B(- |u|^{p-1}u)_{xx}, ~(p>1)$, where the nonlocality enters through…

For spatially one-dimensional run-and-tumble dynamics with mass conservation we develop a coarse phase diagram, that discriminates between global decay to equidistributed constant states, existence of spatially non-trivial waves, and finite…

In this paper we consider the nonlinear dispersive wave equation on the real line, $u_t-u_{txx}+[f(u)]_x-[f(u)]_{xxx}+\bigl[g(u)+\frac{f''(u)}{2}u_x^2\bigr]_x=0$, that for appropriate choices of the functions $f$ and $g$ includes well known…

We are interested in almost global existence cases in the general theory for nonlinear wave equations, which are caused by critical exponents of nonlinear terms. Such situations can be found in only three cases in the theory, cubic terms in…

In this paper we show the existence of ground-state solutions for the energy-critical NLS perturbed with subcritical terms when the space dimension $d\geq4$. However in dimension three, we show that when the perturbation is small enough,…

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 a blow-up solution for the complex-valued semilinear wave equation with power nonlinearity in one space dimension. We first characterize all the solutions of the associated stationary problem as a two-parameter…

We study the blow-up dynamics for the $L^2$-critical focusing half-wave equation on the real line, a nonlocal dispersive PDE arising in various physical models. As in other mass-critical models, the ground state solution becomes a threshold…