The interfaces of an inhomogeneous non-Newtonian polytropic filtration equation with convection

In this paper, we study the finite time blowup and global existence of the interfaces of the inhomogeneous non-Newtonian polytropic filtration equation with convection
$${\rho (x)u_t = (\mid (u^m)_x\mid ^{p-2}(u^m)_x)_x - \lambda (u^q)_x, \qquad x \epsilon \mathbb R, \, t > 0}$$
where ρ(x) tends to zero as
${\mid x \mid \rightarrow \infty}$
. In particular, we give the critical characterization of the left and right interfaces for the typical model case
${\rho (x) = \rho 1/(1 + \mid x \mid)^{K_1}, \quad (x \rightarrow - \infty)}$
and
${\rho (x) = \rho 2/(1 + \mid x \mid)^{K_2}, \quad (x \rightarrow + \infty)}$
, respectively, which extends the corresponding result for the porous medium equations by Ferreira et al. (2006, The interfaces of an inhomogeneous porous medium equation with convection. Comm. Partial Differential Equations, 31, 497–514).