Self-similar behaviour for the equation of fast nonlinear diffusion

We consider the Cauchy problem for the equation ut = ▿ (u-n ▿u) in [RN for the cases N > 2 with 2 /N ⩽ n < 1 and N = 2 with n=1 for which the time-asymptotic behaviour of finite mass solutions has not previously been established. For N > 2 with n = 2 /N the behaviour as t → + ∞ is shown to take an unusual self-similar form. For N > 2 with 2/N < n < 1 and N = 2 with n = 1 solutions extinguish in finite time. In the former case we show that the behaviour close to the extinction time is given by a similarity solution of the second kind and we derive a number of results for the similarity exponent. For N = 2 with n = 1 the solution to the Cauchy problem is not uniquely specified, and we characterize the possible types of solution and establish their behaviour close to extinction. We also indicate how physical considerations can lead to a unique selection from among the available solutions. The limit N→∞is also analysed, illustrating how the various types of asymptotic behaviour arise from the evolution over earlier times.