Long time behavior of solutions for a damped Benjamin–Ono equation

We consider the Benjamin–Ono equation on torus with an additional damping term smallest Fourier modes ( $$\cos $$ and $$\sin ). first prove global well-posedness of this in $$L^2_{r,0}(\mathbb {T})$$ . Then, we describe weak limit points trajectories when time goes to infinity, show that these are strong points. Finally, boundedness higher-order Sobolev norms for equation. Our key tool is Birkhoff map equation, use as adapted nonlinear transform.