Local existence of energy class solutions for the dirac—klein—gordon equations

Abstract

We show a method to eliminate a type of mixed asymptotics in certain free boundary problems, and give two examples of its application. It appears that these problems cannot be handled by the monotonicity formula of Alt et al. [Alt, H. W., Caffarelli, L. A., Friedman, A. (1984). Variational problems with two phases and their free boundaries. Trans. Am. Math. Soc. 282(2):431–461] or by the more recent monotonicity formula of Caffarelli et al. [Caffarelli, L. A., Jerison, D., Kenig, C. E. (2002). Some new monotonicity theorems with applications to free boundary problems. Ann. Math. (2) 155(2):369–404].     

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.     

一类非线性抛物方程解的Blow-up

研究了一类带有齐次Ｄｉｒｉｃｈｌｅｔ边界条件的非线性退化抛物方程．证明了解在有限时间Ｂｌｏｗｕｐ,并给出了Ｂｌｏｗｕｐ时间的一个界．     

Blow-up solutions for a nonlinear wave equation with boundary damping and interior source

In this paper we consider the wave equation with nonlinear damping and source terms. We are interested in the interaction between the boundary damping −|y_t(L,t)|^m−1y_t(L,t) and the interior source |y(t)|^p−1y(t). We find a sufficient condition for obtaining the blow-up solution of the problem. Furthermore, we also obtain that the solution may blow up even if m≥p.     

一类非线性Schr(o)dinger方程的门槛条件

研究一类带调和势并具组合幂非线性项的非线性Schr(o)dinger方程.该方程描述了在磁场势下具有相互吸引的Bose-Einstein凝聚.应用势井方法、凹方法和变分原理,给出了该方程Cauchy问题的整体解和爆破解存在的门槛条件.     

Some Properties of Solutions for a Nonlinear Difusion System(II)

ＳｏｍｅＰｒｏｐｅｒｔｉｅｓｏｆＳｏｌｕｔｉｏｎｓｆｏｒａＮｏｎｌｉｎｅａｒＤｉｆｕｓｉｏｎＳｙｓｔｅｍ（Ｉ）＊）ＷｕＺｈｕｏｑｕｎ（伍卓群）ａｎｄＹｉｎＪｉｎｇｘｕｅ（尹景学）（ＤｅｐａｒｔｍｅｍｔｏｆＭａｔｈｍａｔｉｃｓ，ＪｉｌｉｎＵｎｉｖｅｒｓ...     

带自由边界的调和映射存在性

本文研究带自由边界的调和映射的存在性.用Sack-Uhlenbeck的挠动Morse理论及Micallef和Moore的Morse指标的估计方法,我们得到了从圆盘到Riemann流形的带自由边界的调和映射的存在性.     

退化的非线性抛物方程初值问题解的Blow—up

研究了一类退化的非线性抛物程的Dirichlet问题，通过引入特征函数，利用其性质巧妙地确定“爆破因子”的方法，得到了解爆破的充分条件及解爆破的一个上界。