Long time behavior of solutions of the generalized Benjamin-Bona-Mahony equation in two space dimensions

Abstract

It is shown that the supremum norms of the solutions with small initial data of the generalized Benjamin-Bona-Mahony equation Ut - Llut = (b, Vu) +uP( a, Vu) in x E IR2 , t E JR+, where 0 ::/; b, a E IR2 , p ~ 3 is an integer, decay to zero like t-2 / 3 as t tends to infinity. The proof of this result is based on an analysis of the linear equation Ut- Llut = (b, Vu) which is much more difficult than in the one-dimensional case studied by J. Albert in (1) and (2). Introduction. We consider in this paper the Cauchy problem for the nonlinear dispersive equation