The Second Critical Exponent for a Fast Diffusion Equation with Potential

摘要

This paper considers a fast diffusion equation with potential ut =△um -V(x)um+up inRn×(0,T),where1-2/αm+n＜m≤1,p＞1,n≥2,V(x)～ω/|x|2 withω≥0 as|x|→∞,and α is the positive root of αm(αm+n-2)-ω=0.The critical Fujita exponent was determined as pc =m + 2/αm+n in a previous paper of the authors.In the present paper,we establish the second critical exponent to identify the global and non-global solutions in their co-existence parameter region p＞pc via the critical decay rates of the initial data.With u0(x)～|x|-a as |x| → ∞,it is shown that the second critical exponent a*=2/p-m,independent of the potential parameter ω,is quite different from the situation for the critical exponent Pc.