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Self-similar solutions of a fast diffusion equation that do not conserve mass. (English) Zbl 0845.35057
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 $$n\downarrow 2/m$$.

##### MSC:
 35K65 Degenerate parabolic equations
##### Keywords:
self-similar solutions; phase space techniques