Asymptotics of Solutions for Periodic Problem for the Korteweg-de Vries Equation with Landau Damping, Pumping and Higher Order Convective Non Linearity

We study the periodic problem for the Korteweg–de Vries equation with Landau damping, linear pumping and a higher-order convective nonlinearity wt+wxxx-αwxx=βw+λwx2wxx,x∈Ω,t>0,w(0,x)=ψx,x∈Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{c} w_{t}+w_{xxx}-\alpha w_{xx}=\beta w+\lambda w_{x}^{2}w_{xx},\text { }x\in \Omega ,t>0,\\ w(0,x)=\psi \left( x\right) ,\text { }x\in \Omega , \end{array} \right. \end{aligned}$$\end{document}where, α,β>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ,\beta >0,$$\end{document}λ∈R,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in \mathbb {R},$$\end{document}Ω=-π,π.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega =\left[ -\pi ,\pi \right] .$$\end{document} We assume that the initial data ψx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi \left( x\right) $$\end{document} are 2π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\pi $$\end{document} - periodic. We prove the global existence of solutions and analyze their large-time asymptotics.