On well-posedness and wave operator for the gKdV equation

We consider the generalized Korteweg–de Vries (gKdV) equation , where k>4 is an integer number and μ=±1. We give an alternative proof of the Kenig, Ponce and Vega result in Kenig, Ponce and Vega (1993) [9], , which asserts local and global well-posedness in , with sk=(k−4)/2k. A blow-up alternative in suitable Strichartz-type spaces is also established. The main tool is a new linear estimate. As a consequence, we also construct a wave operator in the critical space , extending the results of Côte (2006) [2].