Self-similar solutions of a fast diffusion equation that do not conserve mass

Abstract

We consider self-similar solutions of the fast diffusion equation $u_t=\nabla\cdot(u^{-n}\nabla u)$ in $(0,\infty)\times\mathbb{R}^N$, for $N\geq3$ and $\frac2N<n<1$, of the form $ u(x,t) = (T-t)^\alpha f\left(\left|x\right|(T-t)^{-\beta}\right). $ Because mass conservation does not hold for these values of $n$, this results in a nonlinear eigenvalue problem for $f$, $\alpha$ and $\beta$. We employ phase plane techniques to prove existence and uniqueness of solutions $(f,\alpha,\beta)$, and we investigate their behaviour when $n\uparrow1$ and when $n\downarrow\frac2N$.