in a cylindrical domain. The main question is that do the weak solutions of (1.1) with fixed initial and boundary values converge in any reasonable sense to the solution of the limit problem as p varies. Apart from mathematical interest, the stability questions is motivated by error analysis in applications: It is desirable that solutions remain stable under small perturbations of the measured parameter p. Equation (1.1) is known as the p-parabolic equation or parabolic pLaplace equation. Sometimes it is also called the non-Newtonian filtration equation which refers to the fact that the equation models the flow of non-Newtonian fluids. For the regularity theory we refer to DiBenedetto’s monograph [4]. See also Chapter 2 of [22]. The equation is singular if 1 < p < 2 and degenerate if p ≥ 2. We shall focus on the degenerate case. The stability turns out to be a rather delicate problem. The main obstruction is that the underlying parabolic Sobolev space changes as p varies and hence the associated energy is not necessarily finite. We give an example of this phenomenon when the lateral boundary of the cylinder is a Cantor type set. In this case, it may also happen that the solutions converge to a solution of a wrong limit problem. These phenomena are already present in the stationary case, see KilpelainenKoskela [8] and Lindqvist [14], but the time dependence offers new challenges. Our main result shows that solutions with varying exponent converge to the solution of the limit problem in the parabolic Sobolev