Related papers: Blow-up for the stochastic nonlinear Schrodinger e…
We consider a non-linear stochastic wave equation driven by space-time white noise in dimension 1. First of all, we state some results about the intermittency of the solution, which have only been carefully studied in some particular cases…
This work is devoted to non-linear stochastic Schr\"odinger equations with multiplicative fractional noise, where the stochastic integral is defined following the Riemann-Stieljes approach of Z\"ahle. Under the assumptions that the initial…
We study behaviors of scalar quantities near the possible blow-up time, which is made of smooth solutions of the Euler equations, Navier-Stokes equations and the surface quasi-geostrophic equations. Integrating the dynamical equations of…
In this paper, we consider the 3d cubic focusing inhomogeneous nonlinear Schr\"{o}dinger equation with a potential $$ iu_{t}+\Delta u-Vu+|x|^{-b}|u|^{2}u=0,\;\;(t,x) \in {{\bf{R}}\times{\bf{R}}^{3}}, $$ where $0<b<1$. We first establish…
We study the blow-up problem of one-dimensional nonlinear heat equations. Our result shows that for a certain class of initial conditions, the solutions blow up in finite time and we characterize the asymptotic dynamics of these solutions.…
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 the present paper, we study the existence and blow-up behavior to the following stochastic non-local reaction-diffusion equation: \begin{equation*} \left\{ \begin{aligned} du(t,x)&=\left[(\Delta+\gamma) u(t,x)+\int_{D}u^{q}(t,y)dy…
We study the singular stochastic wave equation on $\mathbb T^2$, with a cubic nonlinearity and Gaussian rough Mat\'ern forcing (a Fourier multiplier of order $\alpha>0$ applied to space-time white noise) and establish local well-posedness…
Finite time blow-up is shown to occur for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system provided that the mass of the initial condition exceeds an explicit threshold. In the supercritical case, blow-up…
Existence of finite-time blow ups in the classical one-dimensional nonlinear Schr\"odinger equation (NLS) (1) i \partial_t u + u_{x x} + |u|^{2r} u = 0, u(x,0) = u_0(x) has been one of the central problems in the studies of the singularity…
We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2) initial data; in the focusing case we require,…
We construct a new class of multi-solitary wave solutions for the mass critical two dimensional nonlinear Schrodinger equation (NLS). Given any integer K>1, there exists a global (for positive time) solution of (NLS) that decomposes…
We consider the focusing $L^2$-supercritical 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 $d\geq 2, \frac{d}{2d-1} \leq s <1$ and…
This article is concerned with a semilinear time-fractional diffusion equation with a superlinear convex semilinear term in a bounded domain $\Omega$ with the homogeneous Dirichlet, Neumann, Robin boundary conditions and non-negative and…
We study the nonlinear wave equation with a sign-changing potential in any space dimension. If the potential is small and rapidly decaying, then the existence of small-amplitude solutions is driven by the nonlinear term. If the potential…
In this paper, we are concerned with solutions to the Cauchy problem for Chern-Simons-Schr\"odinger equations in the mass supercritical case. First we establish the local well-posedness of solutions in the radial space. Then we consider…
Boundedness and blow-up of solutions for a nonlinear elliptic system arising in probability and stochastic processes
In this paper, we prove existence and uniqueness of energy solutions for nonlinear Schr\"odinger equations with a multiplicative white noise on $R^d$ with $d\le3$. We rely on an exponential trans-form and conserved quantities for existence…
In this paper, blowup phenomenon for the semilinear wave equation with time-dependent speed of propagation and scattering damping is considered under the smallness of initial data. Our result contains small data blowup for sub-Strauss…
We consider the radial focusing energy critical nonlinear wave equation in three spatial dimensions. We establish the stability of the ODE-blowup under random perturbations below the energy space. The argument relies on probabilistic…