English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $n\geq 3$, $0< m<\frac{n-2}{n}$ and $T>0$. We construct positivesolutions to the fast diffusion equation $u_t=\Delta u^m$ in$\mathbb{R}^n\times(0,T)$, which vanish at time $T$. By introducing a scalingparameter $\beta$ inspired by \cite{DKS}, we study the second-order asymptoticsof the self-similar solutions associated with $\beta$ at spatial infinity. Wealso investigate the asymptotic behavior of the solutions to the fast diffusionequation near the vanishing time $T$, provided that the initial value of thesolution is close to the initial value of some self-similar solution andsatisfies some proper decay condition at infinity. Depending on the range ofthe parameter $\beta$, we prove that the rescaled solution converges either toa self-similar profile or to zero as $t\nearrow T$. The former impliesasymptotic stabilization towards a self-similar solution, and the latter is anew vanishing phenomenon even for the case $n\ge3$ and $m=\frac{n-2}{n+2}\,$which corresponds to the Yamabe flow on $\mathbb{R}^n$ with metric$g=u^{\frac{4}{n+2}}dx^2$.

Vanishing time behavior of solutions to the fast diffusion equation