The solution of anisotropic sixth‐order Schrödinger equation

Abstract

This paper studies the local existence of solutions in Sobolev space for anisotropic sixth-order Schrödinger-type equation iu t +Δu+ ∑ d i=1 (a 4 x i u+b 6 x i u)+ c|u| u = 0, x ∈ R n , t ∈ R, 1 ≤ d < n, under the initial conditions u(x, 0) = (x), x ∈ R n . 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 H s 1 , ⃗ H s 2 ,r ⃗ z . KEYWORDS Banach fixed point theorem, global solution, local solution, sixth-order Schrödinger equation, small initial value MSC CLASSIFICATION35Q55 H s p (R n ) < +∞} under the conditions either b ≠ 0 and 5b > max{3a 2 , 0} or b = 0, a < 0, and ||S(t) || E s (S(t) will be defined in Section 2, the same below) and is very small in Zhao 7 or in Zhao and Cui. 8 For the cases n = 2 and d = 1, the problem (1) has a local solution in C([−T, T], H s (R 2 )), also has a global solution in C(R, H s (R 2 )) under the conditions either 5b > 3a max{a, 0} and > 3 or b = 0, a < 0, > 8 3 , and || || H s , and is very small in Guo and Cui. 9 It can be seen from the form of Equation (1) that higher derivatives are not derived in every direction, so it is natural to think of such problems in anisotropic Sobolev spaces. For the case b = 0, in Zhao et al, 10 the existence of local solutions of problem (1) in anisotropic Sobolev space Hgiven, but the global well posedness is not 1868 discussed. In Guo et al, 11 we obtain the global existence in anisotropic Sobolev space C(R, W 3,d 2 (R n )) by energy method for the case b ≠ 0, where W 3,d 2 (R n ) = {u|u, u x ∈ L 2 (R n ), = 1, … , n, u x i x i , u x i x i x i ∈ L 2 (R n ), i = 1, … , d(< n)}. Noting that W 3,d 2 (R n ) is only an integer order Sobolev space, we will study the global solution of (1) in anisotropic fractional order Sobolev space. In this paper, we will firstly give the existence and uniqueness of the local solution for the initial value problem (1) in anisotropic Sobolev spaces H s 1 , ⃗ H s 2 ,r ⃗ z . Secondly, for the cases n = 2 and d = 1, we will give the existence of global solutions in anisotropic Sobolev spaces H s 1 , ⃗ H s 2 ,r ⃗ z . Before our main results are stated, we first introduce some notations.is Banach space with the norm || || H s,r (R n ) = || || L r (R n ) + ||I s || L r (R n ) , where I s = F −1 (| | s̃( )). For simplicity, we define L r