An asymptotic and numerical description of self-similar blow-up in quasilinear parabolic equations

We study the blow-up behaviour of two reaction-diffusion problems with a quasilinear degenerate diffusion and a superlinear reaction. We show that in each case the blow-up is self-similar, in contrast to the linear diffusion limit of each in which the diffusion is only approximately self-similar. We then investigate the limit of the self-similar behaviour and describe the transition from a stable manifold blow-up behaviour (for quasilinear diffusion) to a centre manifold one for the linear diffusion.