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The initial value problem for cubic semilinear Schrödinger equations. (English) Zbl 0870.35095

The author presents local and global existence theorems for cubic semilinear Schrödinger equations. The idea of the proof consists of energy and decay estimates. He remarks that these equations do not allow the classical energy estimates. To avoid this difficulty, he makes strong use of S. Doi’s method for linear Schrödinger type equations [J. Math. Kyoto Univ. 34, 319-328 (1994; Zbl 0807.35026)]. This paper is concerned with the initial value problem for semilinear Schrödinger equations of the form: \[ u_t-i\Delta u=F(u,\nabla u)\quad\text{in }(0,\infty)\times\mathbb{R}^N,\tag{1} \]
\[ u(0,x)= u_0(x)\quad\text{in }\mathbb{R}^N,\tag{2} \] where \(u(t,x)\) is \(\mathbb{C}\)-valued. The author assumes that the nonlinear term \(F(u,q)\in C^\infty(\mathbb{R}^2\times\mathbb{R}^{2N};\mathbb{C})\) satisfies \[ |F(u,q)|\leq C(|u|^p+|q|^p)\quad\text{near }(u,q)=0, \] with some integer \(p\geq 2\), where \(q\in\mathbb{C}^N\) corresponds to \(\nabla u\). The author proves the following
Theorem: Assume \(N\geq 2\) and \(q\geq 3\). Then there exists a sufficiently large integer \(m_1\in\mathbb{N}\) such that for any \(u_0\in H^m\) \((m\geq m_1)\), there exists a time \(T=T(|u_0|_Hm_1)>0\) such that the initial value problem (1)–(2) has a unique solution \(u\in C([0,T);H^m)\).
In the case \(N\geq 3\), \(q\geq 3\) and \(N=2\), \(q\geq 3\) he gives theorems which are finer than the above ones.
Reviewer: A.Tsutsumi (Osaka)

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35G25 Initial value problems for nonlinear higher-order PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)

Citations:

Zbl 0807.35026
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References:

[1] Chihara, H., Local existence for the semilinear Schrbdinger equations in one space dimension, /. Math Kyoto Univ., 34(1994), 353-367. · Zbl 0817.35104
[2] , Local existence for semilinear Schrodinger equations, Math. Japon., 42(1995), 35 52.
[3] , Global existence of small solutions to seminear Schrodinger equations with gauge in- variance, Publ. RIMS, 31(1995), 731-753. · Zbl 0847.35126
[4] , Global existence of small solutions to semilinear Schrodinger equations, Comm. P. D. £.,21 (1996) ,63-78. · Zbl 0843.35111
[5] Doi, S., On the Cauchy problem for Schrodinger type equations and the regularity of the solu- tions, /. Math. Kyoto Univ., 34(1994), 319-328. · Zbl 0807.35026
[6] Hayashi, N., Global existence of small solutions to quadratic nonlinear Schrodinger equations, Comm. P. D. E., 18(1993), 1109-1124. · Zbl 0786.35120
[7] Hayashi, N. and Ozawa, T., Remarks on nonlinear Schrodinger equations in one space dimen- sion, Diff. Integral Eqs., 7(1994), 453-461. · Zbl 0803.35137
[8] , Global, small radially symmetric solutions to nonlinear Schrodinger equations and a gauge transformation, Diff. Integral Eqs., 8(1995), 1061-1072. · Zbl 0823.35157
[9] Hormander, L., The analysis of linear partial differential operators III, Springer-Verlag, Berlin, 1985.
[10] Katayama, S. and Tsutsumi Y., Global existence of solutions for nonlinear Schrodinger equa- tions in one space dimension, Comm. P. D. E., 19(1994), 1971-1997. · Zbl 0832.35130
[11] Kenig, C. E., Ponce, G. and Vega, L., Small solutions to nonlinear Schrodinger equations, Ann. I. H. P. Non. Lin., 10(1993), 255-288. · Zbl 0786.35121
[12] Klainerman, S., Global existence for nonlinear wave equations, Comm. Pure Appl. Math., 33 (1980), 43-101. · Zbl 0405.35056
[13] Klainerman, S., and Ponce, G., Global, small amplitude solutions to nonlinear evolution equa- tions, Comm. Pure Appl. Math., 36(1983), 133-141. · Zbl 0509.35009
[14] Kumano-go, H., Pseudo-differential operators, MIT press, Cambridge, 1981. · Zbl 0179.42201
[15] Mizohata, S., On the Cauchy problem, Academic Press, New York, 1985. · Zbl 0616.35002
[16] Moser, J., A rapidly convergent interaction method and non-linear partial differential equations-I, Ann. Sc. Norm. Sup. Pisa (3), 20(1966), 265-315. · Zbl 0144.18202
[17] Nirenberg, L., On elliptic partial differential equations, Ann. Sc. Norm. Sup. Pisa (3) , 13 (1959), 115-162. · Zbl 0088.07601
[18] Ozawa, T., Remarks on quadratic nonlinear Schrodinger equations, Funkcial. Ekvac., 38 (1995) , 217-232. · Zbl 0842.35060
[19] Shatah, J., Global existence of small solutions to nonlinear evolution equations, /. Diff. Eqs., 46 (1982), 409-425. · Zbl 0518.35046
[20] Soyeur, A., The Cauchy problem for the Ishimori equations, /. Funct. Anal., 105 (1992) , 233-255. · Zbl 0763.35077
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