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)


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


Zbl 0807.35026
Full Text: DOI


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