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Existence of Neumann and singular solutions of the fast diffusion equation

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  • Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$, $n\ge 3$, $0 < m \le \frac{n-2}{n}$, $a_1,a_2,\dots, a_{i_0}\in\Omega$, $\delta_0 = \min_{1 \le i \le i_0} \mbox{dist} (a_i,∂\Omega)$ and let $\Omega_{\delta}=\Omega\setminus\cup_{i=1}^{i_0}B_{\delta}(a_i)$ and $\hat{\Omega}=\Omega\setminus\{a_1\,\dots,a_{i_0}\}$. 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 $a_1,\dots, a_{i_0}$.
    Mathematics Subject Classification: 35A01, 35K67.


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