Related papers: L^2-concentration phenomenon for Zakharov system b…
Recently, we have given the $l^2$ bijectivity for defocusing Ablowitz-Ladik systems in the discrete Sobolev space $l^{2,1}$ by inverse spectral method. Based on these results, the goal of this article is to investigate the long-time…
We construct approximate solutions $ (\psi_*, n_*)$ of the critical 4D Zakharov system which collapse in finite time to a singular renormalization of the solitary bulk solutions $ (\lambda e^{i \theta}W, \lambda^2 W^2)$ . To be precise for…
We consider the $L^2$ critical inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{R}^N$ $$ i \partial_t u +\Delta u +|x|^{-b} |u|^{\frac{4-2b}{N}}u = 0, $$ where $N\geq 1$ and $0<b<2$. We prove that if $u_0\in…
We consider the generalized two-dimensional Zakharov-Kuznetsov equation $u_t+\partial_x \Delta u+\partial_x(u^{k+1})=0$, where $k\geq3$ is an integer number. For $k\geq8$ we prove local well-posedness in the $L^2$-based Sobolev spaces…
In this paper, we study the solutions to the energy-critical quadratic nonlinear Schr\"odinger system in ${\dot H}^1\times{\dot H}^1$, where the sign of its potential energy can not be determined directly. If the initial data ${\rm u}_0$ is…
In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space $\R^3$. The existence of global in time nearby Maxwellian solutions is known from [34] in 2006.…
We consider solutions of the focusing cubic and quintic Gross-Pitaevskii (GP) hierarchies. We identify an observable corresponding to the average energy per particle, and we prove that it is a conserved quantity. We prove that all solutions…
This paper is concerned with the local well-posedness and the precise blow-up scenario for a periodic 2-component \mu-Hunter-Saxton system in Besov spaces. Moreover, we state a new global existence result to the system. Our obtained results…
We prove the small energy scattering for the three-dimensional generalized Zakharov system with radial symmetry based on the idea by Guo and Nakanishi (2014), which treats the usual Zakharov system. For the proof, we use the…
This paper deals with the convergence to 0 of the energy of the solutions of a second order linear coupled system. In order to obtain the energy decay, we exhibit a Liapunov function.
We show global well-posedness in energy norm of the semi-relativistic Schr\"odinger-Poisson system of equations with attractive Coulomb interaction in ${\mathbb R}^3$ in the presence of pseudo-relativistic diffusion. We also discuss…
We prove global existence and scattering for small localized solutions of the Cauchy problem for the Zakharov system in 3 space dimensions. The wave component is shown to decay pointwise at the optimal rate of t^{-1}, whereas the…
This paper is concerned with the attraction-repulsion chemotaxis system with superlinear logistic degradation, \begin{align*} \begin{cases} u_t = \Delta u - \chi \nabla\cdot(u \nabla v) + \xi \nabla\cdot (u \nabla w) + \lambda u - \mu u^k,…
In this paper, we consider the Cauchy problem of the Geng-Xue system with cubic nonlinearity. Firstly, we prove a blow-up criteria in the low besov space. Secondly, we prove the blow-up phenomenon by using the method which does not require…
We consider the Cauchy problem for the energy critical heat equation $$ u_t = \Delta u + |u|^{\frac 4{n-2}}u {{\quad\hbox{in } }} \ {\mathbb R}^n \times (0, T), \quad u(\cdot,0) =u_0 {{\quad\hbox{in } }} {\mathbb R}^n $$ in dimension $n=5$.…
We consider the modified Zakharov-Kuznetsov (mZK) equation in two space dimensions in both focusing and defocusing cases. Using the $I$-method, we prove the global well-posedness of the $H^s$ solutions for $s>\frac{3}{4}$ for any data in…
We consider the nonlinear Schr\"odinger equation$$-\Delta u + V(x)\,u = a\,u^p + \mu u \quad \text{in }\mathbb{R}^n,\qquad \int_{\mathbb{R}^n} u^2 = 1,$$modeling attractive Bose--Einstein condensates. For all dimensions $n\ge 2$ and all…
We investigate the Landau-Coulomb equation and show an explicit blow-down mechanism for a family of initial data that are small-scale, supercritical perturbations of a Maxwellian function. We establish global well-posedness and show that…
Local and global well-posedness results are established for the initial value problem associated to the 1D Zakharov-Rubenchik system. We show that our results are sharp in some situations by proving Ill-posedness results otherwise. The…
In this paper, we study the blow-up radial solution of fully parabolic system with higher dimensional two species Cauchy problem for some initial condition. In addition, we show that the set of positive radial functions in…