  This is the first of two papers devoted to the study of the properties of the
blow-up surface for the $N$ dimensional semilinear wave equation with
subconformal power nonlinearity. In a series of papers, we have clarified the
situation in one space dimension. Our goal here is to extend some of the
properties to higher dimension. In dimension one, an essential tool was to
study the dynamics of the solution in similarity variables, near the set of
non-zero equilibria, which are obtained by a Lorentz transform of the
space-independent solution. As a matter of fact, the main part of this paper is
to study similar objects in higher dimensions. More precisely, near that set of
equilibria, we show that solutions are either non-global, or go to zero, or
converge to some explicit equilibrium. We also show that the first case cannot
occur in the characteristic case, and that only the third possibility occurs in
the non-characteristic case, thanks to the non-degeneracy of the blow-up limit,
another new result in our paper. As a by-product of our techniques, we obtain
the stability of the zero solution.
