The generalized Korteweg–de Vries–Burgers equation in H2(R)H2(R)

The generalized KdV–Burgers equation ut+(δuxx+g(u))x−νuxx+γu=f(x)ut+(δuxx+g(u))x−νuxx+γu=f(x), δ,ν>0,γ≥0δ,ν>0,γ≥0, is considered in this paper. Using the parabolic regularization technique we prove local and global solvability in H2(R)H2(R) of the Cauchy problem for this equation. Several regularity properties of the approximations stay valid for solutions constructed in such a way. Next, for when γ>0γ>0, we study the asymptotic behavior of the corresponding semigroup on H2(R)H2(R), constructing the (H2(R),H3−(R))(H2(R),H3−(R)) global attractor.