Critical convective-type equations on a half-line

We are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for critical convective-type dissipative equationsut+ℕ(u,ux)+(an∂xn+am∂xm)u=0,(x,t)∈ℝ+×ℝ+,u(x,0)=u0(x),x∈ℝ+,∂xj−1u(0,t)=0forj=1,…,m/2, where the constantsan,am∈ℝ,n,mare integers, the nonlinear termℕ(u,ux)depends on the unknown functionuand its derivativeuxand satisfies the estimate|ℕ(u,v)|≤C|u|ρ|v|σwithσ≥0,ρ≥1, such that((n+2)/2n)(σ+ρ−1)=1,ρ≥1,σ∈[0,m). Also we suppose that∫ℝ+xn/2ℕdx=0. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem above-mentioned. We find the main term of the asymptotic representation of solutions in critical case. Also we give some general approach to obtain global existence of solution of initial-boundary value problem in critical convective case and elaborate general sufficient conditions to obtain asymptotic expansion of solution.