Neumann problem for the Korteweg–de Vries equation

AbstractWe consider Neumann initial-boundary value problem for the Korteweg–de Vries equation on a half-line(0.1){ut+λuux+uxxx=0,t>0,x>0,u(x,0)=u0(x),x>0,ux(0,t)=0,t>0. We prove that if the initial data u0∈H10,214∩H21,72 and the norm ‖u0‖H10,214+‖u0‖H21,72⩽ε, where ε>0 is small enough Hps,k={f∈L2;‖f‖Hps,k=‖〈x〉k〈i∂x〉sf‖Lp<∞}, 〈x〉=1+x2 and λ∫0∞xu0(x)dx=λθ<0. Then there exists a unique solution u∈C([0,∞),H21,72)∩L2(0,∞;H22,3) of the initial-boundary value problem (0.1). Moreover there exists a constant C such that the solution has the following asymptoticsu(x,t)=Cθ(1+ηlogt)−1t−23Ai′(xt3)+O(ε2t−23(1+ηlogt)−65) for t→∞ uniformly with respect to x>0, where η=−9θλ∫0∞Ai′2(z)dz and Ai(q) is the Airy functionAi(q)=12πi∫−i∞i∞e−z3+zqdz=1πRe∫0∞e−iξ3+iξqdξ

Initial-boundary value problems for nonlinear pseudoparabolic equations in a critical case

We study nonlinear pseudoparabolic equations, on the half-line in a critical case,$displaylines{ partial _{t}( u-u_{xx}) -alpha u_{xx}=lambda |u| u,quad xin mathbb{R}^{+},; t>0, cr u( 0,x) =u_{0}( x) , quad xin mathbb{R}^{+}, cr u(t,0)=0,}$$where$alpha >0$,$lambda in mathbb{R}$. The aim of this paper is to prove the existence of global solutions to the initial-boundary value problem and to find the main term of the asymptotic representation of solutions

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 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

Benjamin-Ono Equation on a Half-Line

We consider the initial-boundary value problem for Benjamin-Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time