| In 1860, Schrodinger first put forward the concept "Schrodinger equations" in quantum mechanics and since then, the study on Schrodinger equations has never stopped, for the mathematical description of many physical phenomena belongs to the field of Schrodinger equations, such as nonlinear optic, plasma physics, fluid mechanics etc. As for the form of Schrodinger equations, linear Schrodinger equations was gradually replaced by nonlinear Schrodinger equations; as for the methods of solving Schrodinger equations, the modulus estimate of energy, the principle of contraction mapping, Fourier transformation and harmonic analysis are used; as for the space of the solutions, many people have worked on the problem in bounded domain, Euclidean space of dimension n, periodic bounded conditions and mixed regions and they also combined it with the generalization from low dimension to high dimension. On the one hand, the form of the nonlinear term of Schrodinger equations is relatively monotone, on the other hand, it has many wide application in physics, so it's of significance and necessity to establish a further study.At present, nonlinear Schrodinger equations are mainly concerned and there many already known results of scalar nonlinear Schrodinger equations (short for NLS) with the form (see [4,5,6]). When choosing the sign "minus" or in defocusing case, the global existence of the solution can easily be obtained by the modulus estimate of energy. There are more discussion about the situation when choosing the sign "add" or in focusing case. We now know from [6] that when p < 1 + 4/n, the solution ofSchrodinger equations globally exists in H1(Rn), while p≥1 + 4/n, there only exists local solution and that under some appropriate initial conditions the solution may even blow up. It's interesting to investigate the sufficient conditions of existence of global solution when p≥ 1+ 4/n. In [4], Weinstein skillfully got the best constant in Gagliardo-Niremberg inequality using the variational approach and therefore obtained the sufficient condition of the global existence of the solution in critical case p = 1 + 4/n. Using the principle of contraction mapping, [14,15] gruffly showed us the fact that when the initial value is sufficiently small the solution may globally exist. But it's a pity that we don't know exactly how "small". There isn't beautiful results about the blow-upspeed of a blow-up solution, just only the estimate of the lower and super bound of the solution. The construction of the blow-up points can be found in [19]. Later, Bourgain[18] discussed the problem in the periodic bounded area instead of Rn such asT1 = R1/Z1. He chiefly used harmonic analysis to change the procedure of derivation into increasing powers. Recently, H.Takaoka [17] studied the problem with data on R1× T1. He used L4 - L2 Strichartz inequality and got the global existence of the solution in L2(R1×T1). In fact, early in 1983, Fordy and Kulish [8] first studied the generalization of NLS to vector values therefore created matrix nonlinear Schrodinger equations (short for MNLS) and it can be treated as a family of Schrodinger equations. Though the concept "MNLS" appeared early, articles about it are only published recent years and only discuss the global existence of solution and local existence of periodic solution etc in low dimension. In this research paper, we base our study on NLS and generalize some important results about blowing up solution in NLS to MNLS. Moreover, we'd like to study the blow-up speed of solution, since there are many results about blow-up speed obtained in Elliptic equations, hyperbolic equations but not in Schrodinger equations. I've looked through the latest 3 years' "Chinese mathematical digest" and "Mathematics Review" and found there's no paper which is mostly concerned at the study of blow-up speed of NLS although in some references there are some related results implied in them. So the study on blow-up speed is a good subject. But unfortunately, under the restriction of the space dim...
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