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Register a new userAsymptotic behaviour of solutions of the fast diffusion equation near its extinction time
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Kin Ming Hui
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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 $g_{lambda}in C^{infty}(R^nsetminus{0})$ of the elliptic equation $Delta v^m+alpha v+beta xcdot abla v=0$, $v>0$, in $R^nsetminus{0}$, satisfying $displaystylelim_{|x|to 0}|x|^{alpha/beta}g_{lambda}(x)=lambda^{-frac{rho_1}{(1-m)beta}}$. When $beta$ is sufficiently large, we prove the higher order asymptotic behaviour of radially symmetric solutions of the above elliptic equation as $|x|toinfty$. We also obtain an inversion formula for the radially symmetric solution of the above equation. As a consequence we will prove the extinction behaviour of the solution $u$ of the fast diffusion equation $u_t=Delta u^m$ in $R^ntimes (0,T)$ near the extinction time $T>0$.
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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 singular solution $u$ of the above equation that is trapped in between self-similar solutions of the form of $t^{-alpha} f_i(t^{-beta}x)$, $i=1,2$, with initial value $u_0$ satisfying $A_1|x|^{-gamma}le u_0le A_2|x|^{-gamma}$ for some constants $A_2>A_1>0$ and $frac{2}{1-m}<gamma<frac{n-2}{m}$, where $beta:=frac{1}{2-gamma(1-m)}$, $alpha:=frac{2beta-1}{1-m},$ and the self-similar profile $f_i$ satisfies the elliptic equation $$ Delta f^m+alpha f+beta xcdot abla f=0quad mbox{in ${mathbb R}^nsetminus{0}$} $$ with $lim_{|x|to0}|x|^{frac{ alpha}{ beta}}f_i(x)=A_i$ and $lim_{|x|toinfty}|x|^{frac{n-2}{m}}{f_i}(x)= D_{A_i} $ for some constants $D_{A_i}>0$. When $frac{2}{1-m}<gamma<n$, under an integrability condition on the initial value $u_0$ of the singular solution $u$, we prove that the rescaled function $$ tilde u(y,tau):= t^{,alpha} u(t^{,beta} y,t),quad{ tau:=log t}, $$ converges to some self-similar profile $f$ as $tautoinfty$.
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 diffusion equation $u_t=Delta u^m$, $u>0$, in $widehat{Omega}times (0,infty)$ ($widehat{R^n}times (0,infty)$ respectively) which satisfies $u(x,t)toinfty$ as $xto a_i$ for any $t>0$ and $i=1,cdots,i_0$, when $0<m<frac{n-2}{n}$, $ngeq 3$, and the initial value satisfies $0le u_0in L^p_{loc}(2{Omega}setminus{a_1,cdots,a_{i_0}})$ ($u_0in L^p_{loc}(widehat{R^n})$ respectively) for some constant $p>frac{n(1-m)}{2}$ and $u_0(x)ge lambda_i|x-a_i|^{-gamma_i}$ for $xapprox a_i$ and some constants $gamma_i>frac{2}{1-m},lambda_i>0$, for all $i=1,2,dots,i_0$. We also find the blow-up rate of such solutions near the blow-up points $a_1,a_2,dots,a_{i_0}$, and obtain the asymptotic large time behaviour of such singular solutions. More precisely we prove that if $u_0gemu_0$ on $widehat{Omega}$ ($widehat{R^n}$, respectively) for some constant $mu_0>0$ and $gamma_1>frac{n-2}{m}$, then the singular solution $u$ converges locally uniformly on every compact subset of $widehat{Omega}$ (or $widehat{R^n}$ respectively) to infinity as $ttoinfty$. If $u_0gemu_0$ on $widehat{Omega}$ ($widehat{R^n}$, respectively) for some constant $mu_0>0$ and satisfies $lambda_i|x-a_i|^{-gamma_i}le u_0(x)le lambda_i|x-a_i|^{-gamma_i}$ for $xapprox a_i$ and some constants $frac{2}{1-m}<gamma_ilegamma_i<frac{n-2}{m}$, $lambda_i>0$, $lambda_i>0$, $i=1,2,dots,i_0$, we prove that $u$ converges in $C^2(K)$ for any compact subset $K$ of $2{Omega}setminus{a_1,a_2,dots,a_{i_0}}$ (or $widehat{R^n}$ respectively) to a harmonic function as $ttoinfty$.
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 $mathbb{R}^ntimes (0,infty)$. We will also construct initial data such that the corresponding solution of the fast diffusion equation in bounded domain oscillate between infinity and some positive constant as $ttoinfty$.
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 $g=u^{frac{4}{n+2}}dx^2$.
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 $(mathbb{R}^nsetminus{0})times mathbb{R}$ of the form $U_{lambda}(x,t)=e^{-alpha t}f_{lambda}(e^{-beta t}x), xin mathbb{R}^nsetminus{0}, tinmathbb{R},$ where $f_{lambda}$ is a radially symmetric function satisfying $$frac{n-1}{m}Delta f^m+alpha f+beta xcdot abla f=0 text{ in }mathbb{R}^nsetminus{0},$$ with $underset{substack{rto 0}}{lim}frac{r^2f(r)^{1-m}}{log r^{-1}}=frac{2(n-1)(n-2-nm)}{|beta|(1-m)}$ and $underset{substack{rtoinfty}}{lim}r^{frac{n-2}{m}}f(r)=lambda^{frac{2}{1-m}-frac{n-2}{m}}$, for some constant $lambda>0$. As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation $u_t=frac{n-1}{m}Delta u^m$ in $(mathbb{R}^nsetminus{0})times (0,infty)$ with initial value $u_0$ satisfying $f_{lambda_1}(x)le u_0(x)le f_{lambda_2}(x)$, $forall xinmathbb{R}^nsetminus{0}$, which satisfies $U_{lambda_1}(x,t)le u(x,t)le U_{lambda_2}(x,t)$, $forall xin mathbb{R}^nsetminus{0}, tge 0$, for some constants $lambda_1>lambda_2>0$. We also prove the asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $ttoinfty$ when $n=3,4$ and $frac{n-2}{n+2}le m<frac{n-2}{n}$ holds. Asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $ttoinfty$ is also obtained when $3le n<8$, $1-sqrt{2/n}le m<minleft(frac{2(n-2)}{3n},frac{n-2}{n+2}right)$, and $u(x,t)$ is radially symmetric in $xinmathbb{R}^nsetminus{0}$ for any $t>0$ under appropriate conditions on the initial value $u_0$.
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