اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدUniqueness and time oscillating behaviour of finite points blow-up solutions of the fast diffusion equation
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Kin Ming Hui
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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 $ttoinfty$.
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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
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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 singu
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{ in } Omega, end{cases} end{equation*} where $5leq nleq 7$, $Omega=R^n$ or $Omega subset R^n$ is a smooth, bounded domain enjoying special symmetries. We construct type II finite time blow-up solution $u(x,t)$ with the singularity taking place along an $(n-4)$-dimensional {em shrinking sphere} in $Omega$. More precisely, at leading order, the solution $u(x,t)$ is of the sharply scaled form $$u(x,t)approx la^{-1}(t)frac{2sqrt{2}}{1+left|frac{(r,z)-(xi_r(t),xi_z(t))}{la(t)}right|^2}$$ where $r=sqrt{x_1^2+cdots+x_{n-3}^2}$, $z=(x_{n-2},x_{n-1},x_n)$ with $x=(x_1,cdots,x_n)inOmega$. Moreover, the singularity location $$(xi_r(t),xi_z(t))sim (sqrt{2(n-4)(T-t)},z_0)~mbox{ as }~t earrow T,$$ for some fixed $z_0$, and the blow-up rate $$la(t)sim frac{T-t}{|log(T-t)|^2}~mbox{ as }~t earrow T.$$ This is a completely new phenomenon in the parabolic setting.
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