Generalized Korteweg–de Vries equation in H1(R)H1(R)

We consider the generalized KdV equation ut+uxxx+(g(u))x+γu=f(x)ut+uxxx+(g(u))x+γu=f(x), γ⩾0γ⩾0. We prove its local and global solvability in the phase space H1(R)H1(R) and next, when γ>0γ>0, we study asymptotic behavior of solutions constructing a global attractor of the generalized KdV equation in H1(R)H1(R).