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Register a new userCollapsing behaviour of a singular diffusion equation
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Kin Ming Hui
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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 $T=int_{R^2}u_0dx/4pi$.
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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 $u_0$.
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 $u_0(x)approxfrac{2(n-2)}{beta}frac{log |x|}{|x|^2}$ as $|x|toinfty$. We prove that the rescaled solution $widetilde{u}(x,t)=e^{2beta t}u(e^{beta t}x,t)$ of the maximal global solution $u$ of the equation $u_t=Deltalog u$ in $R^ntimes (0,infty)$, $u(x,0)=u_0(x)$ in $R^n$, converges uniformly on every compact subset of $R^n$ and in $L^1(R^n)$ to $psi_{lambda_0}$ as $ttoinfty$. Moreover $|widetilde{u}(cdot,t)-psi_{lambda_0}|_{L^1(R^n)} le e^{-(n-2)beta t}|u_0-psi_{lambda_0}|_{L^1(R^n)}$ for all $tge 0$.
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 $A_1(1+|x|^2)^{alpha_1}le u_0le A_2(1+|x|^2)^{alpha_2}$ in $R^n$ for some constants $1/(1+ u)lealpha_1<1$, $alpha_2gealpha_1$ and $A_2ge A_1= (2alpha_1(1-3)(n+2alpha_1-2))^{-1/(1+ u)}$ where $0<3<1$ is a constant, the global solution $u$ exists and satisfies $A_1(1+|x|^2+b_1t)^{alpha_1}le u(x,t)le A_2(1+|x|^2+b_2t)^{alpha_2}$ in $R^ntimes (0,infty)$ where $b_1=2(n+2alpha_1-2)3$ and $b_2=2n$ if $0<alpha_2le 1$ and $b_2=2(n+2alpha_2-2)$ if $alpha_2>1$. We also find various conditions on the initial value for the solution to extinct in a finite time and obtain the corresponding decay rate of the solution near the extinction time.
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 $hat{Omega}=Omegasetminus{a_1,...,a_{i_0}}$. For any $0<delta<delta_0$ we will prove the existence and uniqueness of positive solution of the Neumann problem for the equation $u_t=Delta u^m$ in $Omega_{delta}times (0,T)$ for some $T>0$. We will prove the existence of singular solutions of this equation in $hat{Omega}times (0,T)$ for some $T>0$ that blow-up at the points $a_1,..., a_{i_0}$.
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 $tautoinfty$.
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