We study the Cauchy problem to the semilinear fourth-order Schrödinger equations: i∂tu+∂x4u=G∂xkuk≤γ,∂xku¯k≤γ,t>0,x∈R,u|t=0=u0∈Hs(R),(4NLS)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} i\partial _t u+\partial _x^4u=G\left( \left\{ \partial _x^{k}u\right\} _{k\le \gamma },\left\{ \partial _x^{k}{\bar{u}}\right\} _{k\le \gamma }\right) , &{} t>0,\ x\in {\mathbb {R}},\\ \ \ \ u|_{t=0}=u_0\in H^s({\mathbb {R}}), \end{array}\right. }\quad \quad (4\mathrm{NLS}) \end{aligned}$$\end{document}where γ∈{1,2,3}\documentclass[12pt]{minimal}
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\begin{document}$$\gamma \in \{1,2,3\}$$\end{document} and the unknown function u=u(t,x)\documentclass[12pt]{minimal}
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\begin{document}$$u=u(t,x)$$\end{document} is complex valued. In this paper, we consider the nonlinearity G of the polynomial G(z)=G(z1,…,z2(γ+1)):=∑m≤|α|≤lCαzα,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} G(z)=G(z_1,\ldots ,z_{2(\gamma +1)}) :=\sum _{m\le |\alpha |\le l}C_{\alpha }z^{\alpha }, \end{aligned}$$\end{document}for z∈C2(γ+1)\documentclass[12pt]{minimal}
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\begin{document}$$z\in {\mathbb {C}}^{2(\gamma +1)}$$\end{document}, where m,l∈N\documentclass[12pt]{minimal}
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\begin{document}$$m,l\in {\mathbb {N}}$$\end{document} with 3≤m≤l\documentclass[12pt]{minimal}
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\begin{document}$$3\le m\le l$$\end{document} and Cα∈C\documentclass[12pt]{minimal}
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\begin{document}$$C_{\alpha }\in {\mathbb {C}}$$\end{document} with α∈(N∪{0})2(γ+1)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \in ({\mathbb {N}}\cup \{0\})^{2(\gamma +1)}$$\end{document} is a constant. The purpose of the present paper is to prove well-posedness of the problem (4NLS) in the lower order Sobolev space Hs(R)\documentclass[12pt]{minimal}
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\begin{document}$$H^s({\mathbb {R}})$$\end{document} 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.
