journal of differential equations 139, 339364 (1997) Global Solutions for the KdV Equation with Unbounded Data Carlos E. Kenig Department of Mathematics, University of Chicago, Chicago, Illinois 60637 Gustavo Ponce Department of Mathematics, University of California, Santa Barbara, California 93106 and Luis Vega Departamento de Matematicas, Universidad del Pais Vasco, Apartado 644, 48080 Bilbao, Spain Received November 18, 1996; revised April 1, 1997 1. INTRODUCTION This paper is concerned with the initial value problem (IVP) associated to the Kortewegde Vries (KdV) equation {  t u + 3 x u +u x u =0, u( x, 0)=u 0 ( x). t >0, x # R, (1.1) The IVP (1.1) has been extensively studied. In particular, its local and global solvability in the classical Sobolev spaces H s ( R), and its weighted versions have recently received a lot of attention, (for a partial list referen- ces see [4]). In the global setting, J. Bourgain [2] showed that (1.1) is globally well posed in u 0 # L 2 ( R). In [4] we simplified this L 2 result, and extended it, in the local setting, to Sobolev spaces with negative index, i.e. s <0. Roughly, the arguments in [2, 4] combine the dispersive part of the equation in (1.1) modeled by the linear part together with the structure of the nonlinear term. Our interest here is on the global solvability of the IVP (1.1) with smooth and unbounded data u 0 . This problem was suggested by a question raised to us by E. Witten [7]. In this regard we find the following result due to A. Menikoff [6]: article no. DE973297 339 0022-039697 25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.