一类Schrdinger方程的Cauchy问题的整体解

The global solutions of Cauchy problem for a class of Schrdinger equations

研究一类在非线性光学中提出的Schrdinger方程的Cauchy问题iut+Δu+|u|p-1u=0;u(x,0)=u0(x),x∈Rn,t≥0的整体解存在性问题,由于此时问题已不再具有正定能量,通过利用Galerkin结合位势井的方法证明了在满足条件1<p<∞,n=1,2;1<p≤(n+2)/(n-2),n≥3,u0(x)∈H1(Rn),0<E(0)<d,I(u)>0或‖u0‖=0时,问题存在整体解u(x,t):u∈W,0≤t<∞且u∈L∞(0,∞;H1(Rn)),ut∈L∞(0,∞;L2(Rn)).结论中扩大了文献[3]中对p的取值范围,从实质上改进和推广了文献[3]的结果。

The existence of the global solutions of Cauchy problem for a class of Schrodinger equations is studied.The class of Schrodinger equations is raised from the nonlinear optics and can be described as iut＋△u＋｜u｜p-1u=0;u（x,0）=uo（x）,x∈R^n,t≥0.By using potential wells combined with the Galerkin method, it is shown that ifp-satisfies 1〈P〈∞,n=1,2;1〈P≤n＋2/n-2,n≥3,uo（x）∈H^1（R^n）,0〈E（0）〈d,I（u）〉0and｜｜uo｜｜=0,then the problem has the global solutions u（ x, t ）：u∈W,0≤t〈∞andu∈L^∞（0,∞;H^1（R^n））,u1∈L^∞（0,∞;L^2（R^n））.The result extended the range ofpin [ 3 ], and generalizes and improves in essential those results in [3 ].