Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space

Abstract

<p style='text-indent:20px;'>For <inline-formula><tex-math id="M1">\begin{document}$ n\ge 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ 0&lt;m&lt;\frac{n-2}{n} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \beta&lt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \alpha = \frac{2\beta}{1-m} $\end{document}</tex-math></inline-formula>, we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in <inline-formula><tex-math id="M5">\begin{document}$ (\mathbb{R}^n\setminus\{0\})\times \mathbb{R} $\end{document}</tex-math></inline-formula> of the form <inline-formula><tex-math id="M6">\begin{document}$ U_{\lambda}(x,t) = e^{-\alpha t}f_{\lambda}(e^{-\beta t}x), x\in \mathbb{R}^n\setminus\{0\}, t\in\mathbb{R}, $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M7">\begin{document}$ f_{\lambda} $\end{document}</tex-math></inline-formula> is a radially symmetric function satisfying</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \frac{n-1}{m}\Delta f^m+\alpha f+\beta x\cdot\nabla f = 0 \text{ in }\mathbb{R}^n\setminus\{0\}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with <inline-formula><tex-math id="M8">\begin{document}$ \underset{\substack{r\to 0}}{\lim}\frac{r^2f(r)^{1-m}}{\log r^{-1}} = \frac{2(n-1)(n-2-nm)}{|\beta|(1-m)} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \underset{\substack{r\to\infty}}{\lim}r^{\frac{n-2}{m}}f(r) = \lambda^{\frac{2}{1-m}-\frac{n-2}{m}} $\end{document}</tex-math></inline-formula>, for some constant <inline-formula><tex-math id="M10">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation <inline-formula><tex-math id="M11">\begin{document}$ u_t = \frac{n-1}{m}\Delta u^m $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M12">\begin{document}$ (\mathbb{R}^n\setminus\{0\})\times (0,\infty) $\end{document}</tex-math></inline-formula> with initial value <inline-formula><tex-math id="M13">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula> satisfying <inline-formula><tex-math id="M14">\begin{document}$ f_{\lambda_1}(x)\le u_0(x)\le f_{\lambda_2}(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M15">\begin{document}$ \forall x\in\mathbb{R}^n\setminus\{0\} $\end{document}</tex-math></inline-formula>, such that the solution <inline-formula><tex-math id="M16">\begin{document}$ u $\end{document}</tex-math></inline-formula> satisfies <inline-formula><tex-math id="M17">\begin{document}$ U_{\lambda_1}(x,t)\le u(x,t)\le U_{\lambda_2}(x,t) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M18">\begin{document}$ \forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0 $\end{document}</tex-math></inline-formula>, for some constants <inline-formula><tex-math id="M19">\begin{document}$ \lambda_1&gt;\lambda_2&gt;0 $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We also prove the asymptotic large time behaviour of such singular solution <inline-formula><tex-math id="M20">\begin{document}$ u $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M21">\begin{document}$ n = 3,4 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M22">\begin{document}$ \frac{n-2}{n+2}\le m&lt;\frac{n-2}{n} $\end{document}</tex-math></inline-formula> holds. Asymptotic large time behaviour of such singular solution <inline-formula><tex-math id="M23">\begin{document}$ u $\end{document}</tex-math></inline-formula> is also obtained when <inline-formula><tex-math id="M24">\begin{document}$ 3\le n&lt;8 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M25">\begin{document}$ 1-\sqrt{2/n}\le m&lt;\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right) $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M26">\begin{document}$ u(x,t) $\end{document}</tex-math></inline-formula> is radially symmetric in <inline-formula><tex-math id="M27">\begin{document}$ x\in\mathbb{R}^n\setminus\{0\} $\end{document}</tex-math></inline-formula> for any <inline-formula><tex-math id="M28">\begin{document}$ t&gt;0 $\end{document}</tex-math></inline-formula> under appropriate conditions on the initial value <inline-formula><tex-math id="M29">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula>.</p>