Global solutions and their asymptotic behavior for Benjamin-Ono-Burgers type equations (Q5933738)

The authors study the initial-value problem for the equation \(u_t+P(u)_x-\nu u_{xx}-Hu_{xx}=0,\) where \(P\) is a smooth function satisfying certain growth conditions and \(H\) is the Hilbert transform. It is proved that the Cauchy problem for this equation is globally well posed in the Sobolev space \(H^1({\mathbb R})\). Moreover, if \(P(u)=cu^{p+1}\) for \(p>2\) and a constant \(c\), the authors consider the large-time behavior of solutions deriving the equality NEWLINE\[NEWLINE\lim_{t\to\infty}\|u(\cdot,t)-w(\cdot,t)\|_{L^2({\mathbb R})} ={c^2\over 4\nu (8\nu\pi)^{1/2}}\left(\int_0^{+\infty} \int_{-\infty}^{+\infty}u^{p+1}(x,\tau) dx d\tau\right)^2.NEWLINE\]NEWLINE Here, \(w(x,t)\) is the solution to the linearized problem (i.e. for \(P\equiv 0\)).