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 + R 1 2 ∂ x 2 + K ) u ( x , t ) = − u x u , where R α ϕ = 1 2 Γ ( α ) sin ( π 2 α ) ∫ 0 + ∞ ϕ ( y ) | x − y | 1 − α d y is the modified Riesz potential and K u = 1 2 π i θ ( x ) ∫ − i ∞ i ∞ e p x p tanh | p | | p | ( u ˆ ( p , t ) − u ( 0 , t ) p ) d p , where 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.