Related papers: Divergence of infinite-variance nonradial solution…

In this work we prove the existence of solution for a class of perturbed fractional Hamiltonian systems given by \begin{eqnarray}\label{eq00} -{_{t}}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) - L(t)u(t) + \nabla W(t,u(t)) = f(t),…

We prove the growth rate of global solutions of the equation $u_t=\Delta u-u^{-\nu}$ in $\R^n\times (0,\infty)$, $u(x,0)=u_0>0$ in $\R^n$, where $\nu>0$ is a constant. More precisely for any $0<u_0\in C(\R^n)$ satisfying…

In this note, we show the existence of a special solution $u$ to defocusing cubic NLS in $3d$, which lives in $H^{s}$ for all $s>0$, but scatters to a linear solution in a very slow way. We prove for this $u$, for all $\epsilon>0$, one has…

This article considers the stochastic partial differential equation \[ \left\{ \begin{array}{l} u_t = \frac{1}{2} u_{xx} + u^\gamma \xi u(0,.) = u_0 \end{array}\right. \] \noindent where $\xi$ is a space / time white noise Gaussian random…

We study positive radial solutions of quasilinear elliptic systems with a gradient term in the form $$ \left\{ \begin{aligned} \Delta_{p} u&=v^{m}|\nabla u|^{\alpha}&&\quad\mbox{ in }\Omega,\\ \Delta_{p} v&=v^{\beta}|\nabla u|^{q}…

In this paper, we consider a blow-up solution $u(t)$ to the $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5)_x=0$, with finite blow-up time $T<+\infty$. We expect to construct a natural extension of $u(t)$ after the blow-up time. To…

For the 3D cubic quasilinear wave system $\square_{c_i} u^i=G^i(u,\partial u,\partial^2u)=\displaystyle\sum_{\substack{0\le|\alpha|,|\beta|,|\gamma|\le1 \\ 1\le j,k,l \le…

We show the existence and stability of ground state solutions (g.s.s.) for $L^2$-critical magnetic nonlinear Schr\"odinger equations (mNLS) for a class of unbounded electromagnetic potentials. We then give non-existence result by…

In this paper, we study the dispersion-managed nonlinear Schr\"odinger (DM-NLS) equation $$ i\partial_t u(t,x)+\gamma(t)\Delta u(t,x)=|u(t,x)|^{\frac4d}u(t,x),\quad x\in\R^d, $$ and the nonlinearity-managed NLS (NM-NLS) equation: $$…

Let $f$ be a continuous and non-decreasing function such that $f>0$ on $(0,\infty)$, $f(0)=0$, $\sup \_{s\geq 1}f(s)/s< \infty$ and let $p$ be a non-negative continuous function. We study the existence and nonexistence of explosive…

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. More precisely, $$ u_t=\nabla\cdot(u\nabla (-\Delta)^{-s}u), \quad \ 0<s<1. $$ The problem is posed in $\{x\in\ren, t\in…

We study radial symmetry of large solutions of the semi-linear elliptic problem \Delta u + \nabla h.\nabla u = f(|x|,u), and we provide sharp conditions under which the problem has a radial solution. The result is independent of the rate of…

In this paper, we prove the global well-posedness of defocusing 3D quadratic nonlinear Schr\"odinger equation \begin{align*} i\partial_t u + \frac12\Delta u = |u| u, \end{align*} in its sharp critical weighted space $\mathcal F \dot…

We study the critical Neumann problem \begin{equation*} \begin{cases} -\Delta u = |u|^{2^*-2}u &\text{in }\Sigma_\omega,\\ \quad\frac{\partial u}{\partial\nu}=0 &\text{on }\partial\Sigma_\omega, \end{cases} \end{equation*} in the unbounded…

We consider the semilinear heat equation in large dimension $d\geq 11$ $$ \partial_t u =\Delta u+|u| ^{p-1}u, \ \ p=2q+1, \ \ q\in \mathbb N $$ on a smooth bounded domain $\Omega\subset \mathbb R^d$ with Dirichlet boundary condition. In the…

We consider the following fractional NLS with focusing inhomogeneous power-type nonlinearity $$i\partial_t u -(-\Delta)^s u + |x|^{-b}|u|^{p-1}u=0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^N,$$ where $N\geq 2$, $1/2<s<1$, $0<b<2s$ and…

We consider the focusing fractional nonlinear Schr\"odinger equation \[ i\partial_t u - (-\Delta)^s u = -|u|^\alpha u, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $s \in (1/2,1)$ and $\alpha>0$. By using localized virial…

In this paper, we consider a class of the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b} |u|^\alpha u = 0, \quad u(0)=u_0 \in H^1(\mathbb{R}^d), \] with $0<b<\min\{2,d\}$ and $\alpha_\star\leq…

In this note we discuss the global dynamics of an integrable nonlocal NLS on $\mathbb{R}$, which has been the object of recent investigation by integrable systems methods. We prove two results which are in striking contrast with the case of…

We study the Calogero--Moser derivative NLS equation $$ i \partial_t u +\partial_{xx} u + (D+|D|)(|u|^2) u =0 $$ posed on the Hardy-Sobolev space $H^s_+(\mathbb{R})$ with suitable $s>0$. By using a Lax pair structure for this $L^2$-critical…