Cauchy problem for a quasilinear parabolic equation with a source term and an inhomogeneous density

The following quasilinear parabolic equation with a source term and an inhomogeneous density is considered:

$rho (x)frac{{partial u}}{{partial t}} = div(u^{m - 1} left| {Du} right|^{lambda - 1} Du) + u^p $

. The conditions on the parameters of the problem are found under which the solution to the Cauchy problem blows up in a finite time. A sharp universal (i.e., independent of the initial function) estimate of the solution near the blowup time is obtained.