LONG-TIME BEHAVIOR FOR THE EQUATION OF FINITE-DEPTH FLUIDS

In this paper we study the Cauchy problem for the generalized equation of finite-depth fluids partial derivative(t)u - G(partial derivative( x)2u) - partial derivative(x) (u(p)/p) = 0, where G(.) is a singular i ntegral, and p is an integer larger than 1. We obtain the long time be havior of the fundamental solution of linear problem, and prove that t he solutions of the nonlinear problem with small initial data for p > 5/2 + square-root 21/2 are decay in time and freely asymptotic to solu tions of the linear problem. In addition we also study some properties of the singular integral G(.) in L(q)(R) with q > 1.