  This paper discusses the generation of multivariate $C^\infty$ functions with
compact small supports by subdivision schemes. Following the construction of
such a univariate function, called \emph{Up-function}, by a non-stationary
scheme based on masks of {spline subdivision schemes} of growing degrees, we
term the multivariate functions we generate Up-like functions. We generate them
by non-stationary schemes based on masks of box-splines of growing supports. To
analyze the convergence and smoothness of these non-stationary schemes, we
develop new tools which apply to a wider class of schemes than the class we
study. With our method for achieving small compact supports, we obtain, in the
univariate case, Up-like functions with supports $[0, 1 +\epsilon ]$ in
comparison to the support $[0, 2] $ of the Up-function. Examples of univariate
and bivariate Up-like functions are given. As in the univariate case, the
construction of Up-like functions can motivate the generation of $C^\infty$
compactly supported wavelets of small support in any dimension.
