2024/02/23 by Yingying Guo, Guo, Yingying, Weikui Ye +1
Mathematics · #35D30 #35G25 #35Q53 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2402.15128
openalex publication_date 2024/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider the Cauchy problem for the b-equation. Firstly, for s>\frac32, if u0(x)∈ Hs(ℝ) and m0(x)=u0(x)-u0xx(x)∈ L1(ℝ), the global solutions of the b-equation is established when b≥1 or b≤1. It's worth noting that our global result is a new result which doesn't need the condition that m0(x) keeps its sign. For s<\frac32, it is shown (see [13]) that the Cauchy problem of the b-equation is ill-posed in Sobolev space Hs(ℝ) when b>1 or b<1. In the present paper, for s=\frac32, we prove that the Cauchy problem of the b-equation is also ill-posed in H\frac32(ℝ) in the sense of norm inflation by constructing a class of special initial data when b≠1.