The global solution of anisotropic fourth-order Schrödinger equation

Abstract

Abstract This paper studies the global existence of solutions in Sobolev space for anisotropic fourth-order Schrödinger type equation: iut+Δu+a∑i=1duxixixixi+b|u|αu=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∈Rn $x\in R^{n}$, 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∈Rn $x\in R^{n}$. 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 Hy→s1,ρHz→s2,r $H_{\vec{y}}^{s_{1},\rho } H_{\vec{z}} ^{s_{2},r}$.

Keywords

• Anisotropic fourth-order Schrödinger equation
• Global solution
• Small initial value
• Banach fixed point theorem