On the BBM-Burgers Equation: Well-posedness, Ill-posedness and Long Period Limit

Abstract: In this work we study a dispersive equation with a dissipative term, the Benjamin-Bona-Mahony-Burgers equation. First we prove that the initial value problem for this equation is well-posed in $H^s(\mathbb{R}),$ for $s\geq 0$ and ill-posed if $s< 0.$ The ill-posedness is in the sense that the flow-map cannot be continuous at the origin from $H^s(\mathbb{R})$ to even $\mathcal{D}'(\mathbb{R}).$ Additionally, we establish an exact theory of convergence of the periodic solutions to the continuous one, in Sobolev spaces, as the period goes to infinity.