GLOBAL SOLUTION AND REGULARIZING PROPERTIES ON A CLASS OF NONLINEAR EVOLUTION EQUATION

In this paper we will consider the equation u(tt) + M([u(t)]) Au + R([ u(t)]) A(alpha)u + N([u(t)]) Bu(t) = 0. where [u(t)] = ((u,u(t)), (Au, u(t)), parallel to A(1/2)u parallel to(2), parallel to A(1/2)u(t) par allel to(2), parallel to Au parallel to(2)). The initial value problem is proved to be locally well posed for initial data taken in D(A(2)) x D(A(3/2)) and globally well posed for small data, in this case we al so show the exponential decay of tile solution as time goes to infinit y. The main result of this paper is to prove that the solution has the smooting effect property on the initial data. This means that, if the initial data belongs to D(A(2)) x D(A(3/2)) then the solution u belon gs to C-infinity(]0, + infinity[; D(A(k))) For All k is an element of N, provided M, N, and R are C-infinity-function. (C) 1996 Academic Pre ss, Inc.