Weighted Sobolev spaces and rapidly decreasing solutions of some nonlinear dispersive wave equations

Abstract

where d is a positive integer, A is a d-dimensional Laplacian, and the subscripts x, t denote partial differentiations. The initial value problems for (1. I)-(1.3) require the asymptotic conditions as x tends to infinity, and one must specify spaces of functions satisfying the desired asymptotic conditions. The usual choices of spaces are the Sobolev spaces Z-P. Well-posedness in HS, of the initial value problems for (1.1~(1.3), was established by many authors [3,5-8, 12, 181. It is of interest to know that Eqs. (l.l)-( 1.3) have solutions decreasing faster than HS convergence as x tends to infinity, in particular, solutions in the Schwartz spaces S for each t provided that their initial data are in S. This is motivated by Tanaka’s result for the Korteweg-de Vries equation [ 131, which establishes the existence of S-solutions by the inverse scattering techniques, and by the fact that Eqs. (l.l)-(1.3) have travelling solutions in S (solitary waves), which may play an important role of asymptotic behavior of solutions as t tends to infinity.