  Although local existence of multidimensional shock waves has been established
in some fundamental references, there are few results on the global existence
of those waves except the ones for the unsteady potential flow equations in
n-dimensional spaces (n > 4) or in special unbounded space-time domains with
non-physical boundary conditions. In this paper, we are concerned with both the
local and global multidimensional conic shock wave problem for the unsteady
potential flow equations when a pointed piston (i.e., the piston degenerates
into a single point at the initial time) or an explosive wave expands fast in
2-D or 3-D static polytropic gas. It is shown that a multidimensional shock
wave solution of such a class of quasilinear hyperbolic problems not only
exists locally, but it also exists globally in the whole space-time and
approaches a self-similar solution as t goes to infinity.
