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 integral, and p is an integer larger than 1. We obtain the long time behavior of the fundamental solution of linear problem, and prove that the 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 solutions of the linear problem. In addition we also study some properties of the singular integral G(.) in L(q)(R) with q > 1.