Nonlinear Analysis 59 (2004) 55–83 www.elsevier.com/locate/na Nonlinear nonlocal Whitham equation on a segment Elena I. Kaikina ∗ Departamento de Ciencias Básicas, InstitutoTecnológico de Morelia,Av.Technologico, 1500, CP 58120, Morelia, Michoacán, México Received 14 November 2003; accepted 15 July 2004 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  u t + uu x + Ku = 0, t>0,x ∈(0,a), u(x, 0) = u 0 (x), x ∈(0,a), u(a,t) = 0, t>0, (1) where the pseudodifferential operator Ku on a segment [0,a] is defined by Ku= 1 2i  i∞ -i∞ e px K(p) ×   u(p,t) - u(0,t) - e -pa u(a,t) p  dp, with a symbol K(p) = C  p  ,  ∈  1, 3 2  , C  is chosen such that ReK(p)> 0 for Rep = 0. We prove that if the initial data u 0 ∈ L ∞ and ‖u 0 ‖ L ∞ <ε, 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/  + O(t -(1+)/) ), uniformly with respect to the spatial variable x ∈ (0,a), where  = e -i/2 cos /2 i  +i∞ 0 e -K(z) dz.  2004 Elsevier Ltd. All rights reserved. ∗ Corresponding author.Tel./fax: +52-4433121570. E-mail address: ekaikina@matmor.unam.mx (E.I. Kaikina). 0362-546X/$ - see front matter  2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.07.002