在這篇論文中，我們首先回顧三篇跟非線性薛丁格方程的爆破問題，質量聚集以及爆破速率有關的文章。我們盡可能的寫下證明的細節，並且修正一些文章中的小錯誤。另外在第三章，我們給出一個在有限時間內會爆破，並且是正能量的解析解，而這是作者所沒有提到的。
在第六章，我們考慮一個雙方程的薛丁格系統，並且此系統的非線性項是臨界但不對稱。我們使用了位力定理(或稱均功定理)以及在遠處衰退的估計來證明系統的解會在有限時間內爆破。緊接著，我們採用了類似擾動理論的方法。我們使用了一維臨界薛丁格方程的基態、時空伸縮、以及擬保角轉換來構造一個特定的爆破解，並且使用固定點定理來證明修正項函數的存在性，然後我們建構出有限時間內會爆破的系統解。

In this thesis, we first review three papers about the blow-up problem, the mass concentration and the blow-up rate of nonlinear Schrödinger equation. We write down the detail of proof as possible as we can, and give some modification. Especially for Chapter 3, we give an explicit formula of a specific blow-up solution whose energy is positive, which does not mentioned by the authors.
Finally, in Chapter 6, we consider a system of two nonlinear Schrödinger equation,
whose nonlinear term is critical but not symmetric. We use the Virial Identity and
the decay estimate to prove that the solution of the system will blow up in finite time. Next, we apply a method which is some what like the perturbation argument and will be elaborated in Chapter 6. We use the ground state solution, the space time scaling, the pseudo-conformal transformation to construct a specific blow-up solution and use the contraction mapping theorem to prove the existence of the correction term. Finally, we construct a finite time blow-up solution to our system.

References
[1] H. Bahouri, J.-Y. Chemin, and R. Danchin. Fourier analysis and nonlinear partial
differential equations, volume 343. Springer, 2011.

[2] H. Brezis and H. Brézis. Functional analysis, Sobolev spaces and partial differential equations, volume 2. Springer, 2011.

[3] T. Cazenave. Semilinear Schrodinger Equations, volume 10. American Mathematical Soc., 2003.

[4] T. Cazenave and F. B. Weissler. The Cauchy problem for the nonlinear
Schrödinger equation in H1. manuscripta mathematica, 61(4):477–494, 1988.

[5] A. Domarkas. Collapse of solutions of a system of nonlinear Schrödinger equations. Lithuanian Mathematical Journal, 31(4):412–417, 1991.

[6] Y. Giga. Solutions for semilinear parabolic equations in Lp and regularity of weak
solutions of the Navier-Stokes system. Journal of differential equations, 62(2):186–212, 1986.

[7] J. Ginibre and G. Velo. On a class of nonlinear Schrödinger equations. I. The
Cauchy problem, general case. Journal of Functional Analysis, 32(1):1–32, 1979.

[8] J. Ginibre and G. Velo. The global Cauchy problem for the non linear Schrödinger equation revisited. In Annales de l’Institut Henri Poincaré C, Analyse non linéaire, volume 2, pages 309–327. Elsevier, 1985.

[9] R. T. Glassey. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. Journal of Mathematical Physics, 18(9):1794–1797, 1977.

[10] F. Ivanauskas and G. Puriuškis. Blow-up of the solution of a nonlinear Schrödinger equation system with periodic boundary conditions. Nonlinear analysis: modelling and control, 18(1):53–65, 2013.

[11] T. Kato. On nonlinear Schrödinger equations. In Annales de l’IHP Physique
théorique, volume 46, pages 113–129, 1987.

[12] S. V. Manakov. On the theory of two-dimensional stationary self-focusing of
electromagnetic waves. Soviet Physics-JETP, 38(2):248–253, 1974.

[13] F. Merle. Construction of solutions with exactly k blow-up points for the
Schrödinger equation with critical nonlinearity. Communications in mathematical physics, 129(2):223–240, 1990.

[14] F. Merle and P. Raphäel. Blow up of the critical norm for some radial L2 su-
per critical nonlinear Schrödinger equations. American journal of mathematics,
130(4):945–978, 2008.

[15] F. Merle and Y. Tsutsumi. L2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity. Journal of differential equations, 84(2):205–214, 1990.

[16] M. Michel Petrovitch. Sur une manière d’étendre le théorème de la moyenne auxéquations différentielles du premier ordre. Mathematische Annalen, 54(3):417–436, 1901.

[17] H. Nawa and M. Tsutsumi. On blow-up for the pseudo-conformally invariant
nonlinear Schrödinger equation. Funkcialaj Ekvacioj, 32(3):417–428, 1989.

[18] T. Ogawa and Y. Tsutsumi. Blow-up of solutions for the nonlinear Schrödinger
equation with quartic potential and periodic boundary condition. In Functional-
Analytic Methods for Partial Differential Equations, pages 236–251. Springer, 1990.

[19] T. Ogawa and Y. Tsutsumi. Blow-up of H1 solution for the nonlinear Schrödinger equation. Journal of Differential Equations, 92(2):317–330, 1991.
[20] T. Ogawa and Y. Tsutsumi. Blow-up of H1 solutions for the one-dimensional
nonlinear Schrödinger equation with critical power nonlinearity. Proceedings of
the American Mathematical Society, 111(2):487–496, 1991.

[21] M. Reed. Abstract non linear wave equations, volume 507. Springer, 2006.

[22] W. A. Strauss. Existence of solitary waves in higher dimensions. Communications in Mathematical Physics, 55(2):149–162, 1977.

[23] W. A. Strauss. Nonlinear wave equations. Number 73. American Mathematical
Soc., 1990.

[24] K. R. Stromberg. An introduction to classical real analysis, volume 376. AmericanMathematical Soc., 2015.

[25] M. Tsutsumi. Nonexistence of global solutions to the Cauchy problem for the
damped nonlinear Schrödinger equations. SIAM journal on mathematical analysis,
15(2):357–366, 1984.

[26] Y. Tsutsumi. Rate of L2 concentration of blow-up solutions for the nonlinear
Schrödinger equation with critical power. Nonlinear Analysis: Theory, Methods
& Applications, 15(8):719–724, 1990.

[27] M. I. Weinstein. Nonlinear Schrödinger equations and sharp interpolation esti-
mates. Communications in Mathematical Physics, 87(4):567–576, 1982.

[28] M. I. Weinstein. On the structure and formation of singularities in solutions to
nonlinear dispersive evolution equations. Communications in Partial Differential
Equations, 11(5):545–565, 1986.

[29] M. I. Weinstein. The nonlinear Schrodinger equation-Singularity formation,
Stability and dispersion, The connection between Infinite and Finite Dimensional
Dynamical Systems. Contemp. Math., 99:213–232, 1989.

[30] F. B. Weissler. Existence and non-existence of global solutions for a semilinear
heat equation. Israel Journal of Mathematics, 38(1):29–40, 1981.

[31] V. Zakharov, V. Sobolev, and V. Synakh. Behavior of light beams in nonlinear
media. Sov. Phys. JETP, 33(1):77–81, 1971.