Nonlinear Differential Equations and Applications (NoDEA) ( IF 1.1 ) Pub Date : 2021-06-15 , DOI: 10.1007/s00030-021-00707-6 Hiroyuki Hirayama , Masahiro Ikeda , Tomoyuki Tanaka
We study the Cauchy problem to the semilinear fourth-order Schrödinger equations:
$$\begin{aligned} {\left\{ \begin{array}{ll} i\partial _t u+\partial _x^4u=G\left( \left\{ \partial _x^{k}u\right\} _{k\le \gamma },\left\{ \partial _x^{k}{\bar{u}}\right\} _{k\le \gamma }\right) , &{} t>0,\ x\in {\mathbb {R}},\\ \ \ \ u|_{t=0}=u_0\in H^s({\mathbb {R}}), \end{array}\right. }\quad \quad (4\mathrm{NLS}) \end{aligned}$$where \(\gamma \in \{1,2,3\}\) and the unknown function \(u=u(t,x)\) is complex valued. In this paper, we consider the nonlinearity G of the polynomial
$$\begin{aligned} G(z)=G(z_1,\ldots ,z_{2(\gamma +1)}) :=\sum _{m\le |\alpha |\le l}C_{\alpha }z^{\alpha }, \end{aligned}$$for \(z\in {\mathbb {C}}^{2(\gamma +1)}\), where \(m,l\in {\mathbb {N}}\) with \(3\le m\le l\) and \(C_{\alpha }\in {\mathbb {C}}\) with \(\alpha \in ({\mathbb {N}}\cup \{0\})^{2(\gamma +1)}\) is a constant. The purpose of the present paper is to prove well-posedness of the problem (4NLS) in the lower order Sobolev space \(H^s({\mathbb {R}})\) or with more general nonlinearities than previous results. Our proof of the main results is based on the contraction mapping principle on a suitable function space employed by Pornnopparath (J Differ Equ, 265:3792–3840, 2018). To obtain the key linear and bilinear estimates, we construct a suitable decomposition of the Duhamel term introduced by Bejenaru et al. (Ann Math 173:1443–1506, 2011). Moreover we discuss scattering of global solutions and the optimality for the regularity of our well-posedness results, namely we prove that the flow map is not smooth in several cases.
中文翻译：
具有三阶导数非线性的四阶薛定谔方程的适定性
我们研究半线性四阶薛定谔方程的柯西问题：
$$\begin{aligned} {\left\{ \begin{array}{ll} i\partial _t u+\partial _x^4u=G\left( \left\{ \partial _x^{k}u\right\ } _{k\le \gamma },\left\{ \partial _x^{k}{\bar{u}}\right\} _{k\le \gamma }\right) , &{} t>0 ,\ x\in {\mathbb {R}},\\ \ \ \ u|_{t=0}=u_0\in H^s({\mathbb {R}}), \end{array}\right . }\quad \quad (4\mathrm{NLS}) \end{aligned}$$其中\(\gamma \in \{1,2,3\}\)和未知函数\(u=u(t,x)\)是复数值。在本文中，我们考虑多项式的非线性G
$$\begin{aligned} G(z)=G(z_1,\ldots ,z_{2(\gamma +1)}) :=\sum _{m\le |\alpha |\le l}C_{\ alpha }z^{\alpha }, \end{aligned}$$对于\(z\in {\mathbb {C}}^{2(\gamma +1)}\)，其中\(m,l\in {\mathbb {N}}\)与\(3\le m \le l\)和\(C_{\alpha }\in {\mathbb {C}}\)与\(\alpha \in ({\mathbb {N}}\cup \{0\})^{2 (\gamma +1)}\)是一个常数。本文的目的是证明问题 (4NLS) 在低阶 Sobolev 空间\(H^s({\mathbb {R}})\)的适定性或者比以前的结果具有更一般的非线性。我们对主要结果的证明是基于 Pornnopparath 所采用的合适函数空间上的收缩映射原理 (J Differ Equ, 265:3792–3840, 2018)。为了获得关键的线性和双线性估计，我们构建了 Bejenaru 等人引入的 Duhamel 项的合适分解。(Ann Math 173:1443–1506, 2011)。此外，我们讨论了全局解的散射和我们适定结果的规律性的最优性，即我们证明了流图在几种情况下是不平滑的。
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