2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/154344We study the Cauchy problem for the dissipative Benjamin-Ono equations $u_t+\H u_{xx}+|D|^αu+uu_x=0$ with $0\leqα\leq 2$. When $0\leqα< 1$, we show the ill-posedness in $H^s(\R)$, $s\in\R$, in the sense that the flow map $u_0\mapsto u$ (if it exists) fails to be $\C^2$ at the origin. For $1<α\leq 2$, we prove the global well-posedness in $H^s(\R)$, $s>-α/4$. It turns out that this index is optimal.Analysis of PDEs35Q55, 35A05, 35M10text