TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 1, January 2014, Pages 505–530 S 0002-9947(2013)05894-1 Article electronically published on July 16, 2013 THE NONLINEAR HEAT EQUATION WITH HIGH ORDER MIXED DERIVATIVES OF THE DIRAC DELTA AS INITIAL VALUES SLIM TAYACHI AND FRED B. WEISSLER Abstract. In this paper we prove local existence of solutions of the nonlinear heat equation u t =Δu + |u| α u, t ∈ (0,T ),x ∈ R N , with initial value u(0) = K∂ 1 ∂ 2 ··· ∂ m δ, 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 H s,q (R N ) with s ≤−2. As an application, we prove the existence of initial values u 0 = λf for which the resulting solution blows up in finite time if λ> 0 is sufficiently small. Here, f satisfies in particular f ∈ C 0 (R N ) ∩ L 1 (R N ) and is anti-symmetric with respect to x 1 ,x 2 , ··· ,x m . Moreover, we require  R N x 1 ··· x m f (x)dx = 0. This extends the known “small lambda” blow up results which require either that  R N f (x)dx = 0 (Dickstein (2006)) or  R N x 1 f (x)dx = 0 (Ghoul (2011), (2012)). 1. Introduction In this paper we study local existence and uniqueness of solutions to the semi- linear heat equation in integral form, (1.1) u(t)=e tΔ u 0 +  t 0 e (t−σ)Δ ( |u(σ)| α u(σ) ) dσ, where e tΔ is the heat semigroup on R N ,α> 0 and u 0 ∈S  (R N ) is a multiple of (1.2) (−1) m ∂ 1 ∂ 2 ··· ∂ m δ, 1 ≤ m ≤ N and δ is the Dirac point mass at the origin. Recall that e tΔ f = G t f , where (1.3) G t (x) = (4πt) −N/2 e − |x| 2 4t ,t> 0,x ∈ R N . For x =(x 1 ,x 2 , ··· ,x N ), we denote ∂ x i by ∂ i , 1 ≤ i ≤ N. We seek a solution u ∈ C ((0,T ]; C 0 (R N )) such that u(t) → u 0 in S  (R N ) as t → 0.C 0 (R N ) denotes the set of continuous functions on R N which tend to zero at infinity and S  (R N ) denotes the space of tempered distributions on R N . It is well known that (1.1) is locally well posed in C 0 (R N ). In particular, given any u 0 ∈ C 0 (R N ), there exist T max (u 0 ) > 0 and a unique continuous solution Received by the editors November 21, 2011 and, in revised form, May 29, 2012. 2010 Mathematics Subject Classification. Primary 35K55, 35A01, 35B44; Secondary 35K57, 35C15. Key words and phrases. Nonlinear heat equation, highly singular initial values, finite time blow–up. c 2013 American Mathematical Society Reverts to public domain 28 years from publication 505 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use