Critical convective-type equations on a half-line

Abstract

We are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for critical convective-type dissipative equations ut+ℕ(u,ux)+(an∂xn+am∂xm)u=0, (x,t)∈ℝ+×ℝ+, u(x,0)=u0(x), x∈ℝ+, ∂xj−1u(0,t)=0 for j=1,…,m/2, where the constants an,am∈ℝ, n, m are integers, the nonlinear term ℕ(u,ux) depends on the unknown function u and its derivative ux and satisfies the estimate |ℕ(u,v)|≤C|u|ρ|v|σ with σ≥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.