This paper deals with the initial value problem for a class of degenerate nonlinear parabolic equations on a bounded domain in R-N for N >= 2 with the Dirichlet boundary condition. The assumptions ensure that u (math) 0 is a stationary solution and its stability is analysed. Amongst other things the results show that, in the case of critical degeneracy, the principle of linearized stability fails for some simple smooth nonlinearities. It is also shown that for levels of degeneracy less than the critical one linearized stability is justified for a broad class of nonlinearities including those for which it fails in the critical case.