ﻻ يوجد ملخص باللغة العربية
Let $0le u_0(x)in L^1(R^2)cap L^{infty}(R^2)$ be such that $u_0(x) =u_0(|x|)$ for all $|x|ge r_1$ and is monotone decreasing for all $|x|ge r_1$ for some constant $r_1>0$ and ${ess}inf_{2{B}_{r_1}(0)}u_0ge{ess} sup_{R^2setminus B_{r_2}(0)}u_0$ for some constant $r_2>r_1$. Then under some mild decay conditions at infinity on the initial value $u_0$ we will extend the result of P. Daskalopoulos, M.A. del Pino and N. Sesum cite{DP2}, cite{DS}, and prove the collapsing behaviour of the maximal solution of the equation $u_t=Deltalog u$ in $R^2times (0,T)$, $u(x,0)=u_0(x)$ in $R^2$, near its extinction time $T=int_{R^2}u_0dx/4pi$.
For $nge 3$, $0<m<frac{n-2}{n}$, $beta<0$ and $alpha=frac{2beta}{1-m}$, we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in $(mathbb{R}^nsetminus{0})times
Let $nge 3$ and $psi_{lambda_0}$ be the radially symmetric solution of $Deltalogpsi+2betapsi+beta xcdot ablapsi=0$ in $R^n$, $psi(0)=lambda_0$, for some constants $lambda_0>0$, $beta>0$. Suppose $u_0ge 0$ satisfies $u_0-psi_{lambda_0}in L^1(R^n)$ and
We prove the growth rate of global solutions of the equation $u_t=Delta u-u^{- u}$ in $R^ntimes (0,infty)$, $u(x,0)=u_0>0$ in $R^n$, where $ u>0$ is a constant. More precisely for any $0<u_0in C(R^n)$ satisfying $A_1(1+|x|^2)^{alpha_1}le u_0le A_2(1+
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
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