اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدLarge time behaviour of higher dimensional logarithmic diffusion equation
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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 $tge 0$.
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