On the local and global existence of solutions to the nonlocal Whitham equation on half-line

We study the following initial-boundary value problem for the nonlocal Whitham equation u_t+N(u)+Ku=0, (x,t) ∈R⁺ ×R⁺, u(x,0)=u¯(x), x∈R⁺, where the nonlinearity is N(u)=u_xu and K is the pseudodifferential operator on the half-line of order α satisfying 1<α<2 and some dissipative conditions. We prove that if the initial data are such that x^δu¯∈L¹, with δ∈(0,½) and the norm ||u¯||X⁺||x^δu¯||(L ¹) is sufficiently small, where X={ψ∈(L¹), ψ´∈L¹;||ψ||x=||ψ||(L¹ )+||ψ_x||(L¹)<∞}, then there exists a unique solution u∈C ([0, +∞); L²)∩C(R⁺, H¹) of the initial-value problem (1), where H^k is the Sobolev space with norm ||φ||(H^k)=||(1-∂₂^x )k/2φ||(L²). We also study large time asymptotics of the solutions