Nonlinear nonlocal Whitham equation on a segment

Abstract

We study global existence and large time asymptotic behavior of solutions to the initial-boundary value problem for the nonlinear nonlocal equation on a segment (1) u t + uu x + K u = 0 , t > 0 , x ∈ ( 0 , a ) , u ( x , 0 ) = u 0 ( x ) , x ∈ ( 0 , a ) , u ( a , t ) = 0 , t > 0 , where the pseudodifferential operator K u on a segment [ 0 , a ] is defined by K u = 1 2 π i ∫ - i ∞ i ∞ e px K ( p ) × u ^ ( p , t ) - u ( 0 , t ) - e - pa u ( a , t ) p d p , with a symbol K ( p ) = C α p α , α ∈ 1 , 3 2 , C α is chosen such that Re K ( p ) > 0 for Re p = 0 . We prove that if the initial data u 0 ∈ L ∞ and ∥ u 0 ∥ L ∞ < ɛ , then there exists a unique solution u ∈ C ( [ 0 , ∞ ) ; L 2 ( 0 , a ) ) of the initial-boundary value problem (0.1). Moreover, there exists a constant A such that the solution has the following large time asymptotics u ( x , t ) = At - 1 / α Λ + O ( t - ( 1 + δ ) / α ) ) , uniformly with respect to the spatial variable x ∈ ( 0 , a ) , where Λ = e - i π α / 2 cos π α / 2 π i ∫ 0 + i ∞ e - K ( z ) d z .