اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدExtinction profile of the logarithmic diffusion equation
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Let $u$ be the solution of $u_t=Deltalog u$ in $R^Ntimes (0,T)$, N=3 or $Nge 5$, with initial value $u_0$ satisfying $B_{k_1}(x,0)le u_0le B_{k_2}(x,0)$ for some constants $k_1>k_2>0$ where $B_k(x,t) =2(N-2)(T-t)_+^{N/(N-2)}/(k+(T-t)_+^{2/(N-2)}|x|^2)$ is the Barenblatt solution for the equation. We prove that the rescaled function $4{u}(x,s)=(T-t)^{-N/(N-2)}u(x/(T-t)^{-1/(N-2)},t)$, $s=-log (T-t)$, converges uniformly on $R^N$ to the rescaled Barenblatt solution $4{B}_{k_0}(x)=2(N-2)/(k_0+|x|^2)$ for some $k_0>0$ as $stoinfty$. We also obtain convergence of the rescaled solution $4{u}(x,s)$ as $stoinfty$ when the initial data satisfies $0le u_0(x)le B_{k_0}(x,0)$ in $R^N$ and $|u_0(x)-B_{k_0}(x,0)|le f(|x|)in L^1(R^N)$ for some constant $k_0>0$ and some radially symmetric function $f$.
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