Global existence for the nonlinear heat equation in the Fujita subcritical case with initial value having zero mean value

In this paper we prove for 1<p<1+2/N+k, where k is an integer in [1, N], the existence of an initial value ψ, odd with respect to the k first coordinates, and with ∫ _{R{double-struck}N}x ₁...x _kψdx ₁...dx _N≠0, such that the resulting solution of u _t-δu=|u| ^p-1u is global. In the case k=1 and 1<p<1+1N+1, it is known that the solution u with the initial value u(0)=λψ blows up in finite time if λ>0 either sufficiently small or sufficiently large. The result in this paper extends a similar result of Cazenave, Dickstein, and Weissler in the case k=0, i.e. with ∫ _{R{double-struck}N}ψ≠0 and 1<p<1+2/N.