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This is the first of two papers devoted to the study of the properties of theblow-up surface for the $N$ dimensional semilinear wave equation withsubconformal power nonlinearity. In a series of papers, we have clarified thesituation in one space dimension. Our goal here is to extend some of theproperties to higher dimension. In dimension one, an essential tool was tostudy the dynamics of the solution in similarity variables, near the set ofnon-zero equilibria, which are obtained by a Lorentz transform of thespace-independent solution. As a matter of fact, the main part of this paper isto study similar objects in higher dimensions. More precisely, near that set ofequilibria, we show that solutions are either non-global, or go to zero, orconverge to some explicit equilibrium. We also show that the first case cannotoccur in the characteristic case, and that only the third possibility occurs inthe 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 obtainthe stability of the zero solution.

Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions