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 N x 1 ⋯ x k ψ d x 1 ⋯ d x N ≠ 0 , such that the resulting solution of u t − Δ u = | u | p − 1 u is global. In the case k = 1 and 1 < p < 1 + 1 N + 1 , it is known that the solution u with the initial value u ( 0 ) = λ ψ 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 N ψ ≠ 0 and 1 < p < 1 + 2 N .