Whitham equation with Landau damping on a half-line

We consider the initial-boundary value problem on a half-line for the evolution equation(∂t+R12∂x2+K)u(x,t)=−uxu, whereRαϕ=12Γ(α)sin(π2α)∫0+∞ϕ(y)|x−y|1−αdy is the modified Riesz potential andKu=12πiθ(x)∫−i∞i∞epxptanh|p||p|(uˆ(p,t)−u(0,t)p)dp, where θ(x)θ(x) is the Heaviside step function. 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.