The second critical exponent for a semilinear nonlocal parabolic equation

This article considers the Cauchy problem for a semilinear nonlocal parabolic equation ut=Δu+(∫RnK(y)uq(y,t)dy)p−1qur+1 in Rn×(0,T)Rn×(0,T), where p,q⩾1p,q⩾1, r⩾0r⩾0 and p+r>1p+r>1. We study the second critical exponent, i.e. describing the critical smallness of initial data required by global solutions (non-global solutions) via the decay rates of the initial data at spatial infinity. Differently from other parabolic equations, the second critical exponent is related to n   in this problem when K∉L1(Rn)K∉L1(Rn).