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

Abstract

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}$$\\begin{aligned} \\left\\{ \\begin{array}{c} w_{t}+w_{xxx}-\\alpha w_{xx}=\\beta w+\\lambda w_{x}^{2}w_{xx},\ ext { }x\\in \\Omega ,t>0,\\\\ w(0,x)=\\psi \\left( x\\right) ,\ ext { }x\\in \\Omega , \\end{array} \\right. \\end{aligned}$$\\end{document}where, alpha ,beta >0,lambda in mathbb {R},Omega =left[ -pi ,pi right] . We assume that the initial data psi left( xright) are 2pi - periodic. We prove the global existence of solutions and analyze their large-time asymptotics.