Global well-posedness for the Cauchy problem of the viscous Degasperis-Procesi equation (Q1034057)

This paper deals with the Cauchy problem of the viscous Degasperis-Procesi (VDP) equation \[ m_{t} + u\, m_{x} + 3\, u_{x}m = m_{xx}, \quad t>0, \quad x \in \mathbb{R}, \] \[ \qquad \qquad \quad \qquad m (0,x) = u_{0} (x) - u_{0,xx} (x), \quad x \in \mathbb{R}, \] where \(m = u - u_{xx}\). Based on the bilinear estimates, the local-posedness of the initial value problem associated with the VDP equation is proved. They construct the energy identity to prove the global existence theorem of the VDP equation. It is shown that there exists a unique global solution of the VDP equation. In addition, the solution is smooth on \((0,+ \infty) \times \mathbb{R}\). This is an interesting paper which is concerned with the effect of viscosity of the VDP equation.