The second critical exponent for a semilinear nonlocal parabolic equation

This article considers the Cauchy problem for a semilinear nonlocal parabolic equation u t = Δ u + ( ∫ R n K ( y ) u q ( y , t ) d y ) p − 1 q u r + 1 in R n × ( 0 , T ) , where p , q ⩾ 1 , r ⩾ 0 and p + 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 ∉ L 1 ( R n ) .