Estimations optimales en temps petit et près de la frontière pour les solutions de l'équation de la chaleur avec données non compatibles

Abstract

In this Note, we prove small time estimates near the boundary for solutions of the linear heat equation: u t = Δ u in (0, ∞) x Ω, u = 0 on ∂Ω, u(0) =in Ω, with non-compatible initial dataε C(Ω), where Ω is a bounded domain of ℝ N. The proofs are primarily based on the maximum principle, via the construction of suitable sub and super-solutions, and on a monotonicity argument. We obtain similar results for the associated inhomogeneous problem. A generalization to a large class of linear parabolic equations is announced. © Académie des Sciences/Elsevier, Paris