## Differential and Integral Equations

### 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 $n\downarrow\frac2N$.

#### Article information

Source
Differential Integral Equations, Volume 8, Number 8 (1995), 2045-2064.

Dates
First available in Project Euclid: 20 May 2013

https://projecteuclid.org/euclid.die/1369056139

Mathematical Reviews number (MathSciNet)
MR1348964

Zentralblatt MATH identifier
0845.35057

Subjects
Primary: 35K55: Nonlinear parabolic equations

#### Citation

Peletier, M. A.; Zhang, Hong Fei. Self-similar solutions of a fast diffusion equation that do not conserve mass. Differential Integral Equations 8 (1995), no. 8, 2045--2064. https://projecteuclid.org/euclid.die/1369056139