Stability and spectra of blow-up in problems with quasi-linear gradient diffusivity

We study a nonlinear diffusion equation which is invariant under a stretching group of transformations and which reduces Do a linear diffusion equation in the limit of sigma --> 0. Wie show that if sigma > 0 then the equation has solutions which form singularities which have a self-similar profile. By considering a nonlinear eigenvalue problem the existence and stability of the self-similar profiles is discussed. The existence of approximately self-similar behaviour is also considered for evolution profiles with several maxima and minima. This behaviour is compared to similar behaviour for the linear diffusion problem. The paper uses a combination of techniques from analysis and the theory of dynamical systems.