Global Existence/nonexistence of Sign‐Changing Solutions to u t = Δ u + | u | p in R d

Consider the parabolic equation u t = Δ u +|   u   | p   in   R d × ( 0 , T ) ,u ( x , 0 ) = Φ ( x ) in R d ,up to a maximal time T = T ∞ , where p > 1. Let p * = 1 + 2/ d . It is a classical result that if p ⩽ p * , then there exist no non‐negative, global solutions to the above equation for any choice of Φ ⩾ 0; that is, necessarily T ∞ < ∞ and the solution blows up in some sense. On the other hand, if p > p * , then there do exist non‐negative, global solutions for appropriate choices of Φ. A recent paper by Zhang seems to be the first to consider the existence of global solutions for the above equation with sign‐changing initial data. He proved that if p ⩽ p * and∫ R dΦ ( x ) d x > 0, then the solution to the above equation is not global. In this paper it is shown that this result continues to hold if∫ R dΦ ( x ) d x = 0, and then it is shown that for each p > 1 and each l ∈ (−∞,0), there exists a Φ satisfying∫ R dΦ ( x ) d x = land such that the corresponding solution to the above equation is global, and there exists a Φ satisfying∫ R dΦ ( x ) d x = land such that the corresponding solution is non‐global. 2000 Mathematics Subject Classification 35K57, 35K15.