Cauchy problem for fast diffusion equation with localized reaction

This paper studies the Cauchy problem for the fast diffusion equation with a localized reaction. We establish the Fujita type theorem to the problem, and then obtain the diffusion-independent blow-up rate for the non-global solutions. Moreover, we prove that the blow-up set for the problem consists of a single point under large initial data. These conclusions are quite different from those for the slow diffusion case.