Blow up rate for a porous medium equation with power nonlinearity

The present paper is concerned with the blow up rate of a solution to the following Cauchy problem ut=Δum+up,(x,t)∈RN×(0,T), with the bounded initial function u(x,0)=u0(x)≥0, where T<∞ is the blow up time. When u0(x) has a small oscillation around a radial function, we establish the growth rate estimate of the form ‖u(⋅,t)‖L∞(RN)≤C(T−t)−1p−1,∀t∈[0,T), where the constant C is independent of t-variable, and the parameters m and p are expected to be 1