Self-similar solutions and asymptotic behaviour for a class of degenerate and singular diffusion equations

In this paper, we study the self-similar solutions and the time-asymptotic behaviour of solutions for a class of degenerate and singular diffusion equations in the form ut = (|(p(u))_x|λ-2(p(u)) _x)_x, -∞0, where λ > 2 is a constant. The existence, uniqueness and regularity for the self-similar solutions are obtained. In particular, the behaviour at two end points is discussed. Based on the monotonicity property of the self-similar solutions and the comparison principle, we also investigate the time convergence of the solution for the Cauchy problem to the corresponding self-similar solution when the initial data have some decay in space variable. © 2007 The Royal Society of Edinburgh.