Related papers: Almost sure local well-posedness for a derivative …

In this paper, we consider the nonlinear Schr\"odinger equation $iu_t +\Delta u= \lambda |u|^{\frac {4} {N-4}} u$ in $\R^N $, $N\ge 5$, with $\lambda \in \C$. We prove local well-posedness (local existence, unconditional uniqueness,…

We establish probabilistic well-posedness results for the subcubic nonlinear wave equation, posed on the domain $B_2\times\mathbb{T}$, with randomly chosen initial data having radial symmetry in the $B_2$ variable, and with vanishing…

In this article, we follow the strategies, listed in \cite{Burq2011} and \cite{OhPo}, in dealing with supercritical cubic and quintic wave equations, we obtain that, the equation \begin{equation*} \left\{ \begin{split}…

We consider the local well-posedness for 3-D quadratic semi-linear wave equations with radial data: $\Box u = a |\partial_t u|^2+b|\nabla_x u|^2$, $u(0,x)=u_0(x)\in H^{s}_{\mathrm{rad}}$, $\partial_t u(0,x)=u_1(x)\in…

We study the Cauchy problem for the quasilinear wave equation $ \partial^2 _t u = u^{2a} \partial^2_x u + F(u) u_x $ with $a \geq 0$ and show a result for the local in time existence under new conditions. In the previous results, it is…

In this paper, we investigate the problem of optimal regularity for derivative semilinear wave equations to be locally well-posed in $H^{s}$ with spatial dimension $n \leq 5$. We show this equation, with power $2\le p\le 1+4/(n-1)$, is…

This paper investigates the local and global well-posedness for the inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation $iu_{t} +\Delta u=\lambda \left|x\right|^{-b} \left|u\right|^{\sigma } u, u(0)=u_{0} \in L^{2}(\mathbb R^{n})$,…

We prove probabilistic well-posedness for a 2D viscous nonlinear wave equation modeling fluid-structure interaction between a 3D incompressible, viscous Stokes flow and nonlinear elastodynamics of a 2D stretched membrane. The focus is on…

We consider the Cauchy problem of fifth order dispersive equations on the torus. We assume that the initial data is sufficiently smooth and the nonlinear term is a polynomial depending on $\partial_x^3 u, \partial_x^2 u, \partial_x u$ and…

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 paper, we prove a sharp local well-posedness result for spherically symmetric solutions to quasilinear wave equations with rough initial data, when the spatial dimension is three or higher. Our approach is based on Morawetz type…

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main…

We consider the periodic fractional nonlinear Schr\"{o}dinger equation $$ iu_t -(-\Delta)^{\frac{s}{2}} u + \mathcal{N}(|u|)u=0, \quad x\in \mathbb{T}^N,\, \, t \in \mathbb R, \, \, s>0, $$ where the nonlinearity term is expressed in two…

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…

We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main…

For $s \in (\frac{1}{2},1]$ we investigate well-posedness of the equation \[ \left ( i \partial_t + (-\Delta)^{s} \right ) u = \left (|D|^{1-2s} |u|^2 \right)\ |D|^{2s-1} u \] under small initial data…

We prove local and global well-posedness for semi-relativistic, nonlinear Schr\"odinger equations $i \partial_t u = \sqrt{-\Delta + m^2} u + F(u)$ with initial data in $H^s(\mathbb{R}^3)$, $s \geq 1/2$. Here $F(u)$ is a critical Hartree…

We establish that the initial value problem for the quadratic non-linear Schr\"odinger equation $$ iu_t - \Delta u = u^2$$ where $u: \R^2 \times \R \to \C$, is locally well-posed in $H^s(\R^2)$ when $s > -1$. The critical exponent for this…

We consider fractional wave equations with exponential or arbitrary polynomial nonlinearities. We prove the global well-posedness on the support of the corresponding Gibbs measures. We provide ill-posedness constructions showing that the…

We investigate the initial value problem for a defocusing nonlinear Schr\"odinger equation with weighted exponential nonlinearity $$ i\partial_t u+\Delta u=\frac{u}{|x|^b}(e^{\alpha|u|^2}-1); \qquad (t,x) \in \mathbb{R}\times\mathbb{R}^2,…