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

Abstract

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 equation (0.1) u t + N ( u , u x ) - u xx + u xxxx = 0 , ( x , t ) ∈ R + × R + , u ( x , 0 ) = u 0 ( x ) , x ∈ R + , ∂ x j - 1 u ( 0 , t ) = 0 for j = 1 , 2 , where the nonlinear term N ( u , u x ) depends on the unknown function u and its derivative u x and satisfy the estimate N ( u , v ) ⩽ C u ρ v σ with σ ⩾ 0 , ρ ⩾ 1 such that ρ + 3 2 σ = 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.