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

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 $$t\rightarrow \infty $$ .