Well-posedness of the fourth-order perturbed Schrödinger type equation in non-isotropic Sobolev spaces

Abstract

In this paper we study the Cauchy problem of the non-isotropically perturbed fourth-order nonlinear Schrödinger type equation: i u t + △ u + a ∑ i = 1 d u x i x i x i x i + g ( x , | u | ) u = 0 ( ( x 1 , x 2 , … , x n ) ∈ R n , t ⩾ 0 ), where a is a real constant, 1 ⩽ d < n is an integer, g ( x , | u | ) u is a nonlinear function which behaves like | u | α u for some constant α > 0 . By using Kato method, we prove that this perturbed fourth-order Schrödinger type equation is locally well-posed with initial data belonging to the non-isotropic Sobolev spaces H x → s 1 H y → s 2 provided that s 1 , s 2 satisfy the conditions: s 1 ⩾ 0 , s 2 ⩾ 0 for 0 < α < 8 2 n − d or 2 s 1 + 4 s 2 > 2 n − d − 8 α for α ⩾ 8 2 n − d with some additional conditions. Furthermore, by using non-isotropic Sobolev inequality and energy method, we obtain some global well-posedness results for initial data belonging to non-isotropic Sobolev spaces H x → 2 H y → 1 .