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We study the asymptotic behaviour of a sequence of stronglydegenerate parabolic equations $\partial_t (r_h u) - {\rm div}(a_h \cdot Du)$ 
with $r_h(x,t) \geq0$ 
, $r_h \in L^{\infty}(\Omega\times (0,T))$ 
.The main problem is the lack of compactness, by-passed via a regularity result.As particular cases, we obtain G-convergence for elliptic operators $(r_h \equiv 0)$ 
,G-convergence for parabolic operators $(r_h \equiv 1)$ 
, singular perturbationsof an elliptic operator $(a_h \equiv a$ 
and $r_h \to r$ 
, possibly $r\equiv 0)$ 
.
