  We study the (n+1)-dimensional generalization of the dispersionless
Kadomtsev-Petviashvili (dKP) equation, a universal equation describing the
propagation of weakly nonlinear, quasi one dimensional waves in n+1 dimensions,
and arising in several physical contexts, like acoustics, plasma physics and
hydrodynamics. For n=2, this equation is integrable, and it has been recently
shown to be a prototype model equation in the description of the two
dimensional wave breaking of localized initial data. We construct an exact
solution of the n+1 dimensional model containing an arbitrary function of one
variable, corresponding to its parabolic invariance, describing waves, constant
on their paraboloidal wave front, breaking simultaneously in all points of it.
Then we use such solution to build a uniform approximation of the solution of
the Cauchy problem, for small and localized initial data, showing that such a
small and localized initial data evolving according to the (n+1)-dimensional
dKP equation break, in the long time regime, if and only if n=1,2,3; i.e., in
physical space. Such a wave breaking takes place, generically, in a point of
the paraboloidal wave front, and the analytic aspects of it are given
explicitly in terms of the small initial data.
