Subcritical Kuramoto–Sivashinsky-type equation on a half-line

In this paper we are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for subcritical Kuramoto–Sivashinsky-type equationequation(0.1)ut+N(u,ux)-uxx+uxxxx=0,(x,t)∈R+×R+,u(x,0)=u0(x),x∈R+,∂xj-1u(0,t)=0forj=1,2,where the nonlinear term N(u,ux)N(u,ux) depends on the unknown function u   and its derivative uxux and satisfy the estimate|N(u,v)|⩽C|u|ρ|v|σN(u,v)⩽Cuρvσwith σ⩾0,ρ⩾1σ⩾0,ρ⩾1 such thatρ+32σ=2-μ,μ>0.The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) in subcritical case, when the nonlinear term has a time decay rate less than that of the linear terms of Eq. (0.1). Also we find the main term of the asymptotic representation of solutions.