  We analyze $p$-Laplace operators with degenerate elliptic coefficients. This
investigation includes Gru\v{s}in type $p$-Laplace operators. We describe a
\emph{separation phenomenon} in elliptic and parabolic $p$-Laplace type
equations, which provides an illuminating illustration of simple jump
discontinuities of the corresponding weak solutions. Interestingly validity of
an isoperimetric inequality for considered setting does not imply continuity of
elliptic equations. On the other hand, we are able to establish global
$L^1$-$L^\infty$-regularization and decay estimates of every mild solution of
the parabolic Gru\v{s}in type $p$-Laplace equation.
