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

Abstract

AbstractWe study the periodic problem for the Korteweg–de Vries equation with Landau damping, linear pumping and a higher-order convective nonlinearity $$\begin{aligned} \left\{ \begin{array}{c} w_{t}+w_{xxx}-\alpha w_{xx}=\beta w+\lambda w_{x}^{2}w_{xx},\text { }x\in \Omega ,t>0,\\ w(0,x)=\psi \left( x\right) ,\text { }x\in \Omega , \end{array} \right. \end{aligned}$$ w t + w xxx - α w xx = β w + λ w x 2 w xx , x ∈ Ω , t > 0 , w ( 0 , x ) = ψ x , x ∈ Ω , where, $$\alpha ,\beta >0,$$ α , β > 0 , $$\lambda \in \mathbb {R},$$ λ ∈ R , $$\Omega =\left[ -\pi ,\pi \right] .$$ Ω = - π , π . We assume that the initial data $$\psi \left( x\right) $$ ψ x are $$2\pi $$ 2 π - periodic. We prove the global existence of solutions and analyze their large-time asymptotics.