The global solution of anisotropic fourth-order Schrödinger equation

Abstract

This paper studies the local existence of solutions in Sobolev space for anisotropic sixth‐order Schrödinger‐type equation iut+normalΔu+∑i=1dfalse(a∂xi4u+b∂xi6ufalse)+cfalse|u|αu=0,x∈Rn,t∈R,1≤d<n, under the initial conditions u(x,0)=φ(x),x∈Rn. In particular, when n=2 and d=1, we consider the global existence of solutions in Sobolev space for anisotropic sixth‐order Schrödinger equation. By using the Banach fixed point theorem, we obtain the existence, the uniqueness, the continuous dependence of the solution on the initial value, and the decay estimate of the global solution about such problems in anisotropic Sobolev spaces Htruey→s1,ρHtruez→s2,r.