Dead-core rate for the fast diffusion equation with strong absorption in higher dimensional cases

摘要

This paper deals with the dead-core rate for the fast diffusion equation with a strong absorption in several space dimensions ut=Δum−up,(x,t)∈Ω×(0,∞),where 0 < p < m < 1 and Ω=B(0,1)={x∈RN:|x|<1} with N ≥ 1. By using the self-similar transformation technique and the Zelenyak method, in higher dimensional radially symmetric cases, we prove that the dead-core rate is not self-similar. Moreover, we also give the precise estimates on the single-point final dead-core profile. Finally, when the absorption term −up is replaced by −a(x,t)up, then we derive that the dead-core rate can turn into the corresponding ODE rate if the coefficient function a(x, t) is a suitable uniformly bounded positive function, which implies that a(x, t) plays an important role in the study of the dead-core rate. The main aim of this paper is to extend the results obtained by Guo et al. (2010) to the higher dimensional radially symmetric case.