Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation with convective nonlinearity

We consider the periodic problem for the nonlinear damped wave equation with pumping and convective nonlinearity vtt+2αvt−vxx=v+∂xvx3,x∈Ω,t>0v(0,x)=ϕ˜x,vt(0,x)=ψ˜x,x∈Ω,where α>0, Ω=−π,π. We study the solutions, which satisfy the periodic boundary conditions vt,x=vt,2π+x for all x∈R and t>0, with the 2π - periodic initial data ϕ˜x and ψ˜(x). Our aim in the present paper is to find the large time asymptotics for solutions to the periodic problem for the nonlinear damped wave equation (1.1) carefully studying the behavior of the first harmonics of the solution and applying the energy type estimates. We prove the following asymptotics for the solutions vt,x=ϕ˜̂0+12αψ˜̂0+ɛ1+2bɛ2t+O1+ɛ2t−1 as t→∞ uniformly with respect to x∈Ω, where ϕ˜̂0=∫Ωϕ˜xdx, ψ˜̂0=∫Ωψ˜xdx, ɛ=|∫Ωeixϕ˜xdx|, b=32α.