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On the Cauchy problem in Besov spaces for a nonlinear Schrödinger equation. (English) Zbl 0961.35148
The author considers the Cauchy problem of the following nonlinear Schrödinger equation: \[ i\partial_tu+\triangle u = \varepsilon |u|^{\alpha}u, \tag{1} \] \[ u(x,0)=u_0(x),\qquad x \in \mathbb{R}^n,\;t \geq 0, \tag{2} \] where \(\varepsilon\) is either \(1\) or \(-1\), and \(n \geq 2.\)
The homogeneous Besov space \(\dot{B}_2^{s_{\alpha},\infty}\) is defined by \[ f(x) \in \dot{B}_2^{s_{\alpha},\infty} \iff \sup_j 2^{2s_{\alpha}j} \int_{2^j}^{2^{j+2}} |\widehat{f}(\xi)|^2 d \xi < + \infty. \] The author proves
Theorem 1. Let \(\alpha>\frac{4}{n}\), \(\alpha \in 2N \backslash \{0\}\), \(u_0 \in \dot{B}_2^{s_{\alpha},\infty}\), such that \(\|u_0\|_{\dot{B}_2^{s_{\alpha},\infty}}<C_0(\alpha,n)\). Then there exists a global solution of (1),(2) such that \[ u(x,t) \in L_t^{\infty}(\dot{B}_2^{s_{\alpha},\infty}), \] \[ u(x,t) @>>t\to 0> u_0 \text{ weakly in the sense of } \sigma({\mathcal S},{\mathcal S''}). \] Moreover, the solution is unique under an additional assumption. That is (1),(2) is well posed.
A solution is called self-similar if it is invariant by the scaling: \[ \begin{aligned} u_0(x) & \longrightarrow u_{0,\lambda}(x)=\lambda^{\frac{2}{\alpha}}u_0(\lambda x) \\ u(x,t) & \longrightarrow u_{\lambda}(x,t)=\lambda^{\frac{2}{\alpha}}u_0(\lambda x,{\lambda}^2 t).\end{aligned} \] Theorem 2. Under the assumptions of Theorem 1, if moreover \(u_0\) is homogeneous of degree \(-\frac{2}{\alpha}\), the solution is self-similar, \[ u(x,t) = \frac{1}{\sqrt{t}^{\frac{2}{\alpha}}}U \Biggl(\frac{x}{\sqrt{t}}\Biggr), \] and its profile is such that \[ U(x) \in \dot{B}_p^{s_{\alpha},q}, \quad \text{with } \frac{2}{p}+\frac{n}{q}=\frac{n}{2}, \quad q \geq 2,\;(p,q) \neq (2, \infty). \] Author’s main motivation of considering (1),(2) in the Besov space is to derive self-similarity of a solution to (1),(2).

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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