Nonlinear Schrdinger equation with combined power-type nonlinearities and harmonic potential

This paper discusses a class of nonlinear Schrdinger equations with combined power-type nonlinearities and harmonic potential. By constructing a variational problem the potential well method is applied. The structure of the potential well and the properties of the depth function are given. The invariance of some sets for the problem is shown. It is proven that, if the initial data are in the potential well or out of it, the solutions will lie in the potential well or lie out of it, respectively. By the convexity method, the sharp condition of the global well-posedness is given.     

具有多个非线性源项的波动方程

利用位势井方法研究在有界区域上具有多个非线性源项的波动方程初边值问题.给出了位势井的结构和位势井深度函数的性质.通过引进位势井族得到了在这些问题的流之下的一些集合不变性以及解的真空隔离,揭示了只要问题的初值属于位势井内或位势井外,则问题在今后所有时间内的解都存在于位势井内或井外,同时存在一个没有解的空间区域.进而给出了解的整体存在和不存在的门槛结果.最后,利用相同的方法讨论了具有临界初始条件的问题.     

基于交叉变分的非线性Klein—Gordon方程解的整体存在和爆破

研究非线性Klein-Gordon方程的初边值问题,运用位势井方法,在E(0)d的情况得到了方程解的整体存在和爆破.在临界能量状态得到了整体解的存在性与不存在性.最后使用凸性方法,得到某些具有高初始能量解的爆破.     

Asymptotic behaviour of solution for fourth order wave equation with dispersive and dissipative terms

This paper studies the initial boundary value problem of fourth order wave equation with dispersive and dissipative terms. By using multiplier method, it is proven that the global strong solution of the problem decays to zero exponentially as the time approaches infinite, under a very simple and mild assumption regarding the nonlinear term.     

四阶具强阻尼非线性波动方程解的整体存在性与不存在性

研究四阶具强阻尼非线性波动方程utt-△u+△2u-α△ut=f(u)的初边值问题.利用位势井方法,得到了问题整体解存在性与不存在性的最佳条件(门槛结果).     

四阶具强阻尼非线性波动方程解的整体存在性与不存在性

研究四阶具强阻尼非线性波动方程utt-△u+△~2u-α△ut=f(u)的初边值问题.利用位势井方法,得到了问题整体解存在性与不存在性的最佳条件(门槛结果).     

Wave equations .and reaction-diffusion equations with several nonlinear source terms

The initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms in a bounded domain is studied by potential well method.The invariance of some sets under the flow of these problems and the vac- uum isolation of solutions are obtained by introducing a family of potential wells.Then the threshold result of global existence and nonexistence of solutions are given.Finally, the problem with critical initial conditions are discussed.     

具色散耗散项的四阶波动方程解的渐近性质

研究具色散和耗散项的四阶波动方程的初边值问题.使用乘子法,证明了问题的整体强解在关于非线性项简单而宽泛的假设下,随时间趋于无穷而衰减到零.     

具阻尼的高维广义Boussinesq方程的Cauchy问题的整体适定性

研究了一类具阻尼的高维广义Boussinesq方程u_(tt)-△u-△u_(tt)+△~2u-k△u_t=△f(u)的Cauchy问题.在没有建立问题局部解存在性理论的情况下,利用位势井方法分析了阻尼系数k与初值及井深之间的关系,得到了整体解存在与不存在的门槛结果.     

非线性Schr\"{o}dinger 方程解的整体存在和爆破的门槛结果

研究了非线性 Schr\"{o}dinger 方程的柯西问题. 通过引进位势井及其外部集合, 得到了解的整体存在性和爆破的门槛结果.