On decay and regularity properties of solutions to the Benjamin-Ono equation

Key Points

• This study aims to investigate the conditions under which solutions to the Benjamin-Ono equation remain in specific weighted Sobolev spaces over time.
• Examined the solution $u(x,t)$ of the Benjamin-Ono equation in weighted Sobolev spaces.
• Considered persistence properties related to initial data condition in $L^2(|x|^{2\beta} dx)$.
• Assessed regularity requirements, specifically $u_0 \in H^{\beta}(\mathbb{R})$.],
• For $\beta < 7/2$, if $u(x,0) \in L^2(|x|^{2\beta} dx)$ and is regular enough, then $u(x,t)$ remains in the same space.
• Demonstrated a direct correlation between initial data and solution persistence in weighted Sobolev spaces.