Classification of positive solutions of heat equation with supercritical absorption

Let q ≥ 1 + 2 N . We prove that any positive solution of (E) ∂tu − ∆u + u = 0 in R × (0,∞) admits an initial trace which is a nonnegative Borel measure, outer regular with respect to the fine topology associated to the Bessel capacity C 2 q ,q in R N (q = q/q − 1)) and absolutely continuous with respect to this capacity. If ν is a nonnegative Borel measure in R with the above properties we construct a positive solution u of (E) with initial trace ν and we prove that this solution is the unique σ-moderate solution of (E) with such an initial trace. Finally we prove that every positive solution of (E) is σ-moderate.