A complete estimate on the localization for a porous medium type equation

This paper is concerned with the porous medium equation $$u_{t}={\rm div}(u^{\sigma} \nabla u) +u^{\beta},\,\,(x, t) \in R^{N} \times (0,T),$$ where σ >  0, β >  σ +  1, and the blowing-up time T <  ∞. It is shown in 5,7] that the solution u(x,t) is localized in the case when the initial function has a compact support. In addition, an estimate on the size of the localization in terms of the initial support and the blowing-up time T is partially derived in 5] if β >  σ +  3. In this paper we give a complete estimate on the localization for all β >  σ +  1.