Global Well-posedness for the Fourth-order Defocusing Cubic Equation with Initial Data Lying in a Critical Sobolev Space

This paper is devoted to studying the Cauchy problem of the fourth-order defocusing, cubic equation iu_t + Δ²u = − |u|²u in critical Sobolev space. We first prove that the problem is locally well-posed in the critical Sobolev space \(\dot{H}^{s_{c}}(\mathbb{R}^{N})\) $${\dot{H}}^{s_c}({\mathbb{R}}^N)$$ , 3 ≤ N ≤ 7. Using the argument of Dodson in [Rev. Mat. Iberoam., 2022, 38(4): 1087–1100], we further prove that the problem is globally well-posed in some critical Sobolev space \(H_{p}^{s}(\mathbb{R}^{N})\) $$H_p^s({\mathbb{R}}^N)$$ for N = 6 and N = 7 with radial initial data.