Self-similar solutions of a fast diffusion equation that do not conserve mass. (English) Zbl 0845.35057

Peletier, M. A.; Zhang, Hongfei

Differ. Integral Equ. 8, No. 8, 2045-2064 (1995).

From the abstract: The authors consider self-similar solutions of the equation \(u_t= \nabla\cdot (u^{- n} \nabla u)\) in \((0, \infty)\times \mathbb{R}^m\), for \(m> 2\), \(m/2< n< 1\), of the form \(u(x, t)= (T- t)^\alpha\cdot f(|x|(T- t)^{- \beta})\). Because mass conservation law does not hold for these values of \(n\), this results in a nonlinear eigenvalue problem for \(f\), \(\alpha\) and \(\beta\). The authors employ phase space techniques to prove existence and uniqueness of solutions \((f, \alpha, \beta)\), and investigate their behaviour when \(n\uparrow 1\) and \(n\downarrow 2/m\).

Reviewer: G.Sviridyuk (Chelyabinsk)

MSC:

35K65

Degenerate parabolic equations

Keywords:

self-similar solutions; phase space techniques