ترغب بنشر مسار تعليمي؟ اضغط هنا

Asymptotic large time behavior of singular solutions of the fast diffusion equation

182   0   0.0 ( 0 )
 نشر من قبل Soojung Kim
 تاريخ النشر 2015
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

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 $tautoinfty$.

قيم البحث

اقرأ أيضاً

190 - Kin Ming Hui , Soojung Kim 2018
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 stud y 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 $g=u^{frac{4}{n+2}}dx^2$.
72 - Kin Ming Hui 2014
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 $T>0$.
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 $hat{Omega}=Omegasetmi nus{a_1,...,a_{i_0}}$. 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 $a_1,..., a_{i_0}$.
107 - Kin Ming Hui , Jinwan Park 2020
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 $OmegasubsetR^n$ 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 sion 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 $ttoinfty$.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا