STABILITY AND SPECTRA OF BLOW-UP IN PROBLEMS WITH QUASI-LINEAR GRADIENT DIFFUSIVITY

We study a nonlinear diffusion equation which is invariant under a str etching group of transformations and which reduces Do a linear diffusi on equation in the limit of sigma --> 0. Wie show that if sigma > 0 th en the equation has solutions which form singularities which have a se lf-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 o f dynamical systems.