Related papers: The Defocusing Energy-Critical Wave Equation with …

We study the cubic defocusing nonlinear Schr\"odinger equation on $\mathbb{R}^4$ with supercritical initial data. For randomized initial data in $H^s(\mathbb{R}^4)$, we prove almost sure local wellposedness for $\frac{1}{7} < s < 1$ and…

In this article we study the defocusing energy-critical nonlinear wave equation on $\mathbb{R}^4$ with scaling supercritical data. We prove almost sure scattering for randomized initial data in $H^s(\mathbb{R}^4) \times…

In any dimension $n \geq 3$, we show that spherically symmetric bounded energy solutions of the defocusing energy-critical non-linear Schr\"odinger equation $i u_t + \Delta u = |u|^{\frac{4}{n-2}} u$ in $\R \times \R^n$ exist globally and…

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space ($3\leq p<5$) whose initial data are radial and come with a finite energy. In…

Thanks to an approach inspired from Burq-Lebeau \cite{bule}, we prove stochastic versions of Strichartz estimates for Schr\"odinger with harmonic potential. As a consequence, we show that the nonlinear Schr\"odinger equation with quadratic…

This work investigates the semilinear wave equation featuring the displacement dependent term $\sigma(u)\partial_t u $ and nonlinearity $f(u)$. By developing refined space-time a priori estimates under extended ranges of the nonlinearity…

We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H 1 -critical semilinear wave equation on a smooth bounded 2D domain {\Omega}. First, we prove an appropriate Strichartz type…

We consider the focusing cubic nonlinear Schr\"odinger equation \begin{align}\label{CNLSS} i\partial_t U+\Delta U=-|U|^2U\quad\text{on $\mathbb{R}^2\times\mathbb{T}$}.\tag{3NLS} \end{align} Different from the 3D Euclidean case, the…

The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schr\"odinger type equation. Local…

We consider the wave equation with an energy supercritical focusing nonlinearity in general odd dimensions. We prove that any radial solution that remains bounded in the critical Sobolev space is global and scatters to a linear solution.

In this paper we prove global well-posedness and modified scattering for the massive Maxwell-Klein-Gordon equation in the Coulomb gauge on $\mathbb{R}^{1+d}$ $(d \geq 4)$ for data with small critical Sobolev norm. This extends to the…

We consider the wave and Klein-Gordon equations on the real hyperbolic space $\mathbb{H}^{n}$ ($n \geq2$) in a framework based on weak-$L^{p}$ spaces. First, we establish dispersive estimates on Lorentz spaces in the context of…

In this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension $3$. We prove that the set of initial data for which solutions scatter to an unstable excited state $(\phi, 0)$ forms a finite…

In this work we consider a semi-linear energy critical wave equation in ${\mathbb R}^d$ ($3\leq d \leq 5$) \[ \partial_t^2 u - \Delta u = \pm \phi(x) |u|^{4/(d-2)} u, \qquad (x,t)\in {\mathbb R}^d \times {\mathbb R} \] with initial data…

Results of Struwe, Grillakis, Struwe-Shatah, Kapitanski, Bahouri-Shatah, Bahouri-G\'erard and Nakanishi have established global wellposedness, regularity, and scattering in the energy class for the energy-critical nonlinear wave equation…

We consider the energy-critical defocusing nonlinear wave equation (NLW) on $\mathbb{R}^d$, $d=4$ and $5$. We prove almost sure global existence and uniqueness for NLW with rough random initial data in $H^s(\mathbb{R}^d)\times…

We consider the Cauchy problem associated with the modified Zakharov-Kuznetsov equation over $\mathbb{R}^2$. Taking into consideration the associated dispersive effects, we introduce, for $s,a\ge 0$, a two-parameter space…

We consider the cubic non-linear Schr\"odinger equation on general closed (compact without boundary) Riemannian surfaces. The problem is known to be locally well-posed in $H^s(M)$ for $s>1/2$. Global well-posedness for $s\geq 1$ follows…

We continue the study of the Dirichlet boundary value problem of nonlinear wave equation with radial data in the exterior $\Omega = \mathbb{R}^3\backslash \bar{B}(0,1)$. We combine the distorted Fourier truncation method in…

We study the generalized Hartree equation, which is a nonlinear Schr\"odinger-type equation with a nonlocal potential $iu_t + \Delta u + (|x|^{-b} \ast |u|^p)|u|^{p-2}u=0, x \in \mathbb{R}^N$.We establish the local well-posedness at the…