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Existence and large time behaviour of finite points blow-up solutions of the fast diffusion equation

Research paper by Kin Ming Hui, Sunghoon Kim

Indexed on: 18 Jul '18Published on: 17 Jul '18Published in: Calculus of Variations and Partial Differential Equations



Abstract

Let \(\varOmega \subset {\mathbb {R}}^n\) be a smooth bounded domain and let \(a_1,a_2,\ldots ,a_{i_0}\in \varOmega \), \(\widehat{\varOmega }=\varOmega \setminus \{a_1,a_2,\ldots ,a_{i_0}\}\) and \(\widehat{R^n}={\mathbb {R}}^n\setminus \{a_1,a_2,\ldots ,a_{i_0}\}\). We prove the existence of solution u of the fast diffusion equation \(u_t=\varDelta u^m\), \(u>0\), in \(\widehat{\varOmega }\times (0,\infty )\) (\(\widehat{R^n}\times (0,\infty )\) respectively) which satisfies \(u(x,t)\rightarrow \infty \) as \(x\rightarrow a_i\) for any \(t>0\) and \(i=1,\ldots ,i_0\), when \(0<m<\frac{n-2}{n}\), \(n\ge 3\), and the initial value satisfies \(0\le u_0\in L^p_{loc}(\overline{\varOmega }\setminus \{a_1,\ldots ,a_{i_0}\})\) (\(u_0\in L^p_{loc}(\widehat{R^n})\) respectively) for some constant \(p>\frac{n(1-m)}{2}\) and \(u_0(x)\ge \lambda _i|x-a_i|^{-\gamma _i}\) for \(x\approx a_i\) and some constants \(\gamma _i>\frac{2}{1-m},\lambda _i>0\), for all \(i=1,2,\ldots ,i_0\). We also find the blow-up rate of such solutions near the blow-up points \(a_1,a_2,\ldots ,a_{i_0}\), and obtain the asymptotic large time behaviour of such singular solutions. More precisely we prove that if \(u_0\ge \mu _0\) on \(\widehat{\varOmega }\) (\(\widehat{R^n}\), respectively) for some constant \(\mu _0>0\) and \(\gamma _1>\frac{n-2}{m}\), then the singular solution u converges locally uniformly on every compact subset of \(\widehat{\varOmega }\) (or \(\widehat{R^n}\) respectively) to infinity as \(t\rightarrow \infty \). If \(u_0\ge \mu _0\) on \(\widehat{\varOmega }\) (\(\widehat{R^n}\), respectively) for some constant \(\mu _0>0\) and satisfies \(\lambda _i|x-a_i|^{-\gamma _i}\le u_0(x)\le \lambda _i'|x-a_i|^{-\gamma _i'}\) for \(x\approx a_i\) and some constants \(\frac{2}{1-m}<\gamma _i\le \gamma _i'<\frac{n-2}{m}\), \(\lambda _i>0\), \(\lambda _i'>0\), \(i=1,2,\ldots ,i_0\), we prove that u converges in \(C^2(K)\) for any compact subset K of \(\overline{\varOmega }\setminus \{a_1,a_2,\ldots ,a_{i_0}\}\) (or \(\widehat{R^n}\) respectively) to a harmonic function as \(t\rightarrow \infty \).