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

Let Omega subset of R-n be a smooth bounded domain and let , and . We prove the existence of solution u of the fast diffusion equation , , in ( respectively) which satisfies as for any and , when , , and the initial value satisfies ( respectively) for some constant and for and some constants , for all . We also find the blow-up rate of such solutions near the blow-up points , and obtain the asymptotic large time behaviour of such singular solutions. More precisely we prove that if on (, respectively) for some constant and , then the singular solution u converges locally uniformly on every compact subset of (or respectively) to infinity as . If on (, respectively) for some constant and satisfies for and some constants , , , , we prove that u converges in for any compact subset K of (or respectively) to a harmonic function as t -> infinity.