On the Cauchy problem of fourth-order nonlinear Schrödinger equations. (English) Zbl 1115.35121
The paper deals with the Cauchy problem for the fourth-order semilinear Schrödinger equation
\[
\begin{cases} i\partial_tu+a\Delta u +b\Delta^2u=\pm u^p,& (x,t)\in \mathbb R^{n+1},\\ u(0,x)=u_0(x),& x\in \mathbb R^{n}, \end{cases}
\]
where \(p\geq 1\) is a positive integer, \(1\leq n\leq 4,\) \(a\) and \(b\) are real constants with \(b\neq0\) and the initial data \(u_0(x)\) belong to a suitable Besov space. It is proved that for any \(q\in [{{4(p^2-1)}\over{(4-n)p+4+n}},\infty],\) the above Cauchy problem is locally well-posed in \(\dot B^{s_p}_{2,q}\) and \(\dot B^s_{2,q}\) where \(s_p=n/2-4/(p-1)\) and \(s>s_p;\) and almost globally well-posed in these spaces for any \(q\in[1,\infty]\) and small initial data. Global well-posedness is obtained when \(a=0\) for small initial data.
Reviewer: Dian K. Palagachev (Bari)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35R25 | Ill-posed problems for PDEs |
| 35B35 | Stability in context of PDEs |
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