arXiv:1002.4737v5 [math.AP] 23 Aug 2010 Local and global properties of solutions of heat equation with superlinear absorption Tai Nguyen Phuoc Laurent V´ eron Laboratoire de Math´ ematiques et Physique Th´ eorique, Universit´ e Fran¸ cois Rabelais, Tours, FRANCE Abstract We study the limit, when k →∞ of the solutions of ∂ t u − Δu + f (u)=0 in R N × (0, ∞) with initial data kδ, when f is a positive superlinear increasing function. We prove that there exist essentially three types of possible behaviour according f -1 and F -1/2 belong or not to L 1 (1, ∞), where F (t)=  t 0 f (s)ds. We use these results for providing a new and more general construction of the initial trace and some uniqueness and non-uniqueness results for solutions with unbounded initial data. 1991 Mathematics Subject Classification. 35K58; 35K91; 35K15. Key words. Heat equation; singularities; Borel measures; initial trace. Contents 1 Introduction 2 2 Isolated singularities 8 3 About uniqueness 20 4 Initial trace 25 4.1 The regular part of the initial trace ................... 25 4.2 The Keller-Osserman condition holds .................. 27 4.3 The Keller-Osserman condition does not hold ............. 35 1