Complete Classification of the Positive Solutions of Heat Equation with Super Critical Absorption
Konstantinos T. Gkikas, Лаурент Верон · Advanced Nonlinear Studies · 2014
Abstract Let . We prove that any positive solution of (E) ∂ t u − Δu + u q = 0 in ℝ N × (0, ∞) admits an initial trace which is a nonnegative Borel measure, outer regular with respect to the fine topology associated to the Bessel capacity in ℝ N (qʹ = q/(q − 1)) and absolutely continuous with respect to this capacity. If ν is a nonnegative Borel measure in ℝ N 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.