文章基本信息

标题：The global solution of anisotropic fourth-order Schrödinger equation
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作者：Hailing Su ; Cuihua Guo
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2019
卷号：2019
期号：1
页码：1-17
DOI：10.1186/s13662-019-2118-0
出版社：Hindawi Publishing Corporation
摘要：This paper studies the global existence of solutions in Sobolev space for anisotropic fourth-order Schrödinger type equation: i u t + Δ u + a ∑ i = 1 d u x i x i x i x i + b | u | α u = 0 $iu_{t}+\Delta u+a\sum_{i=1}^{d}u_{x_{i} x_{i} x_{i} x_{i}}+b|u|^{ \alpha }u=0$ , x ∈ R n $x\in R^{n}$ , t ∈ R $t\in R$ , 1 ≤ d < n $1\leq d< n$ under the initial conditions: u ( x , 0 ) = φ ( x ) $u(x,0)=\varphi (x)$ , x ∈ R n $x\in R^{n}$ . By using the Banach fixed point theorem, we obtain the existence, the uniqueness, the continuous dependence and the decay estimate of the solution on the initial value in anisotropic Sobolev spaces H y → s 1 , ρ H z → s 2 , r $H_{\vec{y}}^{s_),\rho } H_{\vec{z}} ^{s_,,r}$ .
关键词：Anisotropic fourth-order Schrödinger equation ; Global solution ; Small initial value ; Banach fixed point theorem