ﻻ يوجد ملخص باللغة العربية
Let $ngeq 3$, $0< m<frac{n-2}{n}$ and $T>0$. We construct positive solutions to the fast diffusion equation $u_t=Delta u^m$ in $mathbb{R}^ntimes(0,T)$, which vanish at time $T$. By introducing a scaling parameter $beta$ inspired by cite{DKS}, we study the second-order asymptotics of the self-similar solutions associated with $beta$ at spatial infinity. We also investigate the asymptotic behavior of the solutions to the fast diffusion equation near the vanishing time $T$, provided that the initial value of the solution is close to the initial value of some self-similar solution and satisfies some proper decay condition at infinity. Depending on the range of the parameter $beta$, we prove that the rescaled solution converges either to a self-similar profile or to zero as $t earrow T$. The former implies asymptotic stabilization towards a self-similar solution, and the latter is a new vanishing phenomenon even for the case $nge3$ and $m=frac{n-2}{n+2},$ which corresponds to the Yamabe flow on $mathbb{R}^n$ with metric $g=u^{frac{4}{n+2}}dx^2$.
We study the asymptotic large time behavior of singular solutions of the fast diffusion equation $u_t=Delta u^m$ in $({mathbb R}^nsetminus{0})times(0,infty)$ in the subcritical case $0<m<frac{n-2}{n}$, $nge3$. Firstly, we prove the existence of singu
Let $nge 3$, $0<m<frac{n-2}{n}$, $rho_1>0$, $betagefrac{mrho_1}{n-2-nm}$ and $alpha=frac{2beta+rho_1}{1-m}$. For any $lambda>0$, we will prove the existence and uniqueness (for $betagefrac{rho_1}{n-2-nm}$) of radially symmetric singular solution $g_{
Let $Omega$ be a smooth bounded domain in $R^n$, $nge 3$, $0<mlefrac{n-2}{n}$, $a_1,a_2,..., a_{i_0}inOmega$, $delta_0=min_{1le ile i_0}{dist }(a_i,1Omega)$ and let $Omega_{delta}=Omegasetminuscup_{i=1}^{i_0}B_{delta}(a_i)$ and $hat{Omega}=Omegasetmi
Let $OmegasubsetR^n$ be a smooth bounded domain and let $a_1,a_2,dots,a_{i_0}inOmega$, $widehat{Omega}=Omegasetminus{a_1,a_2,dots,a_{i_0}}$ and $widehat{R^n}=R^nsetminus{a_1,a_2,dots,a_{i_0}}$. We prove the existence of solution $u$ of the fast diffu
Let $nge 3$ and $0<m<frac{n-2}{n}$. We will extend the results of J.L. Vazquez and M. Winkler and prove the uniqueness of finite points blow-up solutions of the fast diffusion equation $u_t=Delta u^m$ in both bounded domains and $mathbb{R}^ntimes (0,