  We consider the homogenization problem for general porous medium type
equations of the form $u_t=\D f(x,\frac{x}{\ve}, u)$. The pressure function
$f(x,y,\cdot)$ may be of two different types. In the type~1 case,
$f(x,y,\cdot)$ is a general strictly increasing function; this is a mildly
degenerate case. In the type~2 case, $f(x,y,\cdot)$ has the form
$h(x,y)F(u)+S(x,y)$, where $F(u)$ is just a nondecreasing function; this is a
strongly degenerate case. We address the initial-boundary value problem for a
general, bounded or unbounded, domain $\Om$, with null (or, more generally,
steady) pressure condition on the boundary. The homogenization is carried out
in the general context of ergodic algebras. As far as the authors know,
homogenization of such degenerate quasilinear parabolic equations is addressed
here for the first time. We also review the existence and stability theory for
such equations and establish new results needed for the homogenization
analysis. Further, we include some new results on algebras with mean value,
specially a new criterion establishing the null measure of level sets of
elements of the algebra, which is useful in connection with the homogenization
of porous medium type equations in the type~2 case.
