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

Miao Chen , Hua Wang , Xiaohua Yao

Frontiers of Mathematics ›› 2024, Vol. 20 ›› Issue (3) : 547 -580.

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Frontiers of Mathematics ›› 2024, Vol. 20 ›› Issue (3) : 547 -580. DOI: 10.1007/s11464-023-0135-5
Research Article

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

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Abstract

This paper is devoted to studying the Cauchy problem of the fourth-order defocusing, cubic equation iut + Δ2u = − |u|2u in critical Sobolev space. We first prove that the problem is locally well-posed in the critical Sobolev space

H˙sc(RN)
, 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
Hps(RN)
for N = 6 and N = 7 with radial initial data.

Keywords

Global well-posedness / fourth-order cubic equation / critical Sobolev space

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Miao Chen, Hua Wang, Xiaohua Yao. Global Well-posedness for the Fourth-order Defocusing Cubic Equation with Initial Data Lying in a Critical Sobolev Space. Frontiers of Mathematics, 2024, 20(3): 547-580 DOI:10.1007/s11464-023-0135-5

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