Abstract
We find a continuum of extinction rates for solutions $u(y,\tau)\ge 0$ ofthe fast diffusion equation$u_\tau=\Delta u^m$ in a subrange of exponents $m\in (0,1)$. The equation isposed in $R^n$ for times up tothe extinction time $T>0$. The rates take the form$\|u(\cdot,\tau)\|_\infty$ ~ $(T-\tau)^\theta$ for a wholeinterval of $\theta>0$. These extinction rates depend explicitly on thespatial decay rates of initial data.
Highlights
We consider the Cauchy problem for the fast diffusion equation: uτ = ∆(um/m), u(y, 0) = u0(y) ≥ 0, y ∈ Rn, τ ∈ (0, T ), y ∈ Rn, (1.1)where m ∈ (0, 1) and T > 0
It is readily seen that the diffusion coefficient c(u) = um−1 → ∞ as u → 0 if m < 1, the name Fast Diffusion Equation (but notice that c(u) → 0 as u → ∞)
We will always work in this range, m < mc, and consider solutions which vanish in a finite time
Summary
The solutions (1.3) play a similar role for m < mc but their basin of attraction may be much smaller This is precisely described in [4], with results on the basin of attraction of the family of generalized Barenblatt solutions; it establishes the optimal rates of convergence of the solutions of (1.1) towards a unique attracting limit state in that family. All of these solutions will have a decay rate near extinction of the form u(·, τ ) ∞ = O((T − τ )nβ), and it is clear that nβ > 1/(1 − m).
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