矩阵非线性薛定谔方程初值问题解的爆破

THE BLOWING UP OF SOLUTIONS TO THE INITIAL VALUE PROBLEM FOR MATRIX NONLINEAR SCHRDINGER EQUATIONS

考虑矩阵非线性薛定谔方程初值问题解的局部存在性及解的爆破问题 ,并且给出了在 H1( Rn)中方程 Bt=i(ΔB+ 2 BB* B) ( n≥ 2 )的解于有限时间内爆破的充分条件 .如果爆破现象出现 ,那么解的某些Lp 范数也在此有限时间内爆破 ,从而可将一般具有形式 iut=-Δu-| u| p-1u( p=3)

This paper considers the local existence and the blowing up of solutions to the Cauchy problem (IVP) for matrix nonlinear Schrdinger equations of the form B t=i(ΔB+2BB *B) in H 1(R n) with n≥2. With this kind of nonlinear term, several sufficient conditions for the blowing up of solutions in H 1(R n) are obtained, and some other L pnorm of a solution also blow up. Therefore, the well-known result for ordinary nonlinear Schrdinger equations with the form iu t=-Δu- |u| p-1u(p=3) can be generalized to matrix nonlinear Schrdinger equations.