A nonlinear wave equation in a time dependent domain

in a noncylindrical open domain Q C B = W x (0, T) (0 < T < + co). By G(t,) (0 < to < T) we denote the intersection of Q with the hyperplane P, = {(x, t) E [Wnfl, t = t,,}. We will also set G(O) = Q n P, and Q(T) = Q n Pr . Let r(t) = Z2(t) and Z = UOGtGT r(t). Z is the lateral boundary of Q. As it should be expected, the position of Z with respect to the light cone plays an important role in the determination of well posed problems for the equations (1). Therefore, throughout this paper we shall assume that Z is an n-dimensional manifold of class Cl. This is more than we need for our results, but this assumption will simplify our discussion. For p E Z we will denote by