On the concentration properties for the nonlinear Schrödinger equation with a Stark potential

In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.     

ON THE CONCENTRATION PROPERTIES FOR THE NONLINEAR SCHRDINGER EQUATION WITH A STARK POTENTIAL

In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.     

Asymptotical Behaviors for Neumann Boundary Problem with Singular Data

In this paper, we will analyze the blow-up behaviors for solutions to the Laplacian equation with exponential Neumann boundary condition. In particular, the boundary value is with a kind of singular data. We show a Brezis–Merle type concentration-compactness theorem, calculate the blow up value at the blow-up point, and give a point-wise estimate for the profile of the solution sequence at the blow-up point.     

Profiles of blow-up solutions for the Gross-Pitaevskii equation

This paper is concerned with the blow-up solutions of the Cauchy problem for Gross-Pitaevskii equation.In terms of Merle and Raphёel＇s arguments as well as Carles＇ transformation,the limiting profiles of blow-up solutions are obtained.In addition,the nonexistence of a strong limit at the blow-up time and the existence of L2 profile outside the blow-up point for the blow-up solutions are obtained.     

Uniform Blow-up Behavior for Degenerate and Singular Parab olic Equations

This paper deals with the degenerate and singular parabolic equations coupled via nonlinear nonlocal reactions, subject to zero-Dirichlet boundary conditions. After giving the existence and uniqueness of local classical nonnegative solutions, we show critical blow-up exponents for the solutions of the system. Moreover, uniform blow-up behaviors near the blow-up time are obtained for simultaneous blow-up solutions, divided into four subcases.     

Blowing up of solutions to the Cauchy problem for the generalized Zakharov system with combined power-type nonlinearities

This paper deals with blowing up of solutions to the Cauchy problem for a class of general- ized Zakharov system with combined power-type nonlinearities in two and three space dimensions. On the one hand, for c0 = +∞ we obtain two finite time blow-up results of solutions to the aforementioned system. One is obtained under the condition α≥ 0 and 1 + 4/N ≤ p N +2/N-2 or α 0 and 1 p 1 + 4/N (N = 2, 3); the other is established under the condition N = 3, 1 p N +2/N-2 and α(p-3) ≥ 0. On the other hand, for c0 +∞ and α(p-3) ≥ 0, we prove a blow-up result for solutions with negative energy to the Zakharov system under study.     

BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR?DINGER EQUATIONS IN THE RADIAL CASE

We consider the blow-up solutions to the following coupled nonlinear Schrodinger equations■.On the basis of the conservation of mass and energy,we establish two sufficient conditions to obtain the existence of a blow-up for radially symmetric solutions.These results improve the blow-up result of Li and Wu [10] by dropping the hypothesis of finite variance((|x|u₀,|x|v₀) ∈L²(R^N) × L²(R^N)).     

REGULARITY OF WEAK SOLUTIONS TO MAGNETO-MICROPOLAR FLUID EQUATIONS

In this article,we study the regularity of weak solutions and the blow-up criteria for smooth solutions to the magneto-micropolar fluid equations in R~3.We obtain the classical blow-up criteria for smooth solutions(u,ω,b),i.e.,u ∈ L q(0,T;L p(R 3)) for 2 q + 3 p ≤ 1 with 3p≤∞,u ∈ C([0,T);L 3(R 3)) or u ∈L q(0,T;L p) for 3 2p≤∞ satisfying 2 q + 3p≤2.Moreover,our results indicate that the regularity of weak solutions is dominated by the velocity u of the fluid.In the end-point case p = ∞,the blow-up criteria can be extended to more general spaces u∈ L~1(0,T;B_(∞,∞)~0(R~3)).     

Blow-up Sets to a Coupled Heat System

This paper deals with a heat system coupled via local and localized sources subject to null Dirichlet boundary conditions. In a previous paper of the authors, a complete result on the multiple blow-up rates was obtained. In the present paper, we continue to consider the blow-up sets to the system via a complete classification for the nonlinear parameters. That is the discussion on single point versus total blow-up of the solutions. It is mentioned that due to the influence of the localized sources, there is some substantial difficulty to be overcomed there to deal with the single point blow-up of the solutions.     

Convergence of global attractors of a 2D non-Newtonian system to the global attractor of the 2D Navier-Stokes system

This paper discusses the relation between the long-time dynamics of solutions of the two-dimensional(2D) incompressible non-Newtonian fluid system and the 2D Navier-Stokes system.We first show that the solutions of the non-Newtonian fluid system converge to the solutions of the Navier-Stokes system in the energy norm.Then we establish that the global attractors {AHε} 0<ε≤1 of the non-Newtonian fluid system converge to the global attractor AH0 of the Navier-Stokes system as → 0.We also construct the minimal limit AHmin of the global attractors {AHε}0<ε≤1 as ε→ 0 and prove that AHmin is a strictly invariant and connected set.     

The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system

The blow-up solutions of the Cauchy problem for the Davey-Stewartson system, which is a model equation in the theory of shallow water waves, are investigated. Firstly, the existence of the ground state for the system derives the best constant of a Gagliardo-Nirenberg type inequality and the variational character of the ground state. Secondly, the blow-up threshold of the Davey-Stewartson system is developed in R³. Thirdly, the mass concentration is established for all the blow-up solutions of the system in R². Finally, the existence of the minimal blow-up solutions in R² is constructed by using the pseudo-conformal invariance. The profile of the minimal blow-up solutions as t→T (blow-up time) is in detail investigated in terms of the ground state.     

Limiting profile of blow-up solutions for the Gross-Pitaevskii equation

This paper is concerned with the blow-up solutions of the Gross-Pitaevskii equation. Using the concentration compact principle and the variational characterization of the corresponding ground state, we obtain the limiting profile of blow-up solutions with critical mass in the corresponding weighted energy space. Moreover, we extend this result to small super-critical mass case by the variational methods and scaling technique. This work was supported by National Natural Science Foundation of China (Grant No. 10771151) and Scientific Research Fund of Sichuan Provincial Education Department (Grant No. 2006A068)     

Long time dynamics of the 3D radial NLS with the combined terms

In this paper,we study the scattering and blow-up dichotomy result of the radial solution to nonlinear Schrodinger equation(NLS) with the combined terms iu_t+△u=-|u|~4u+|4|~(p-1)u,1+4/3p5 in energy space H~1(R~3).The threshold energy is the energy of the ground state W of the focusing,energy critical NLS,which means that the subcritical perturbation does not affect the determination of threshold,but affects the scattering and blow-up dichotomy result with subcritical threshold energy.This extends algebraic perturbation in a previous work of Miao,Xu and Zhao[Comm.Math.Phys.,318,767-808(2013)]to all mass supercritical,energy subcritical perturbation.     

几类具Pucci算子的椭圆方程组解的存在性

研究完全非线性椭圆方程组解的存在性问题,其中ΩR~n,n≥2是有界光滑区域,—Μ_(λ,Λ)~+为具参数0<λ≤Λ的Pucci算子.首先,对f_i,i=1,2为一致有界函数的情形,证明了此方程组存在有界非负解.其次,当{f_1,f_2}是拟增的,且方程组存在有序上、下解时,利用上、下解方法,并结合增算子的不动点定理证明了此方程组存在最大非负解和最小非负解.当{f_1,f_2}是拟减或混拟单调时,使用Schauder不动点定理证明了此方程组至少存在一个非负解.针对此方程组中f_i,i=1,2的某些特殊形式,证明了相应方程组正解的存在性.最后给出了应用实例.     

Davey-Stewartson系统粗糙爆破解的极限行为

研究了Davey-Stewartson系统(简记为D-S系统)粗糙爆破解的动力学性质.所谓粗糙爆破解即为正则性为H~s(s1)的爆破解,此时D-S系统粗糙解不再满足能量守恒率.利用I-方法与Profile分解理论,得到了D-S系统粗糙爆破解在H~s(R~2)(其中ss_0,且s_0≤(1+11~(1/2))/5≈0.8633)中的极限行为,包括L~2强极限的不存在性与L~2集中性质以及极限图景.     

Blow-up for a degenerate parabolic equation with nonlocal source and nonlocal boundary

In this article, we investigate the blow-up properties of the positive solutions for a doubly degenerate parabolic equation with nonlocal source and nonlocal boundary condition. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we give the precise blow-up rate estimate and the uniform blow-up estimate for the blow-up solution.     

Blow-up properties for a degenerate parabolic system with nonlinear localized sources

This paper deals with blow-up properties for a degenerate parabolic system with nonlinear localized sources subject to the homogeneous Dirichlet boundary conditions. The main aim of this paper is to study the blow-up rate estimate and the uniform blow-up profile of the blow-up solution. Our conclusions extend the results of [L.L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources, J. Math. Anal. Appl. 324 (2006) 304-320]. At the end, the blow-up set and blow up rate with respect to the radial variable is considered when the domain Ω is a ball.     

Construction of minimal mass blow-up solutions to rough nonlinear Schrödinger equations

We study the focusing mass-critical rough nonlinear Schrödinger equations, where the stochastic integration is taken in the sense of controlled rough path. In both dimensions one and two, the minimal mass blow-up solutions are constructed, which behave asymptotically like the pseudo-conformal blow-up solutions near the blow-up time. Furthermore, the global well-posedness is obtained if the mass of initial data is below that of the ground state. These results yield that the mass of ground state is exactly the threshold of global well-posedness and blow-up in the stochastic focusing mass-critical case. Similar results are also obtained for a class of nonlinear Schrödinger equations with lower order perturbations, particularly in the absence of the standard pseudo-conformal symmetry and the conservation law of energy.