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In this paper we prove local existence of solutions of the nonlinear heat equation ut = Δu+ |u|αu, t ∈ (0, T ), x ∈ RN , with initial value u(0) = K∂1∂2 · · · ∂mδ, K = 0, m ∈ {1, 2, · · · , N}, 0 < α < 2/(N + m) and δ is the Dirac distribution. In particular, this gives a local existence result with an initial value in a high order negative Sobolev space Hs,q(RN ) with s ≤ −2. As an application, we prove the existence of initial values u0 = λf for which the resulting solution blows up in finite time if λ > 0 is sufficiently small. Here, f satisfies in particular f ∈ C0(R ) ∩ L1(RN ) and is anti-symmetric with respect to x1, x2, · · · , xm. Moreover, we require ∫ RN x1 · · ·xmf(x)dx = 0. This extends the known “small lambda” blow up results which require either that ∫ RN f(x)dx = 0 (Dickstein (2006)) or ∫ RN x1f(x)dx = 0 (Ghoul (2011), (2012)).

The nonlinear heat equation with high order mixed derivatives of the Dirac delta as initial values