Prevention of blowup via Neumann heat kernel

Abstract

Consider the heat equation ut − ∆u = 0 on a bounded C 2 domain Ω in R n (n ≥ 2) with any positive initial data. If a superlinear radiation law ∂u ∂n = u q with q > 1 is imposed on a partial boundary Γ1 ⊆ ∂Ω which has a positive surface area, then it has been known that the solution u blows up in finite time. However, if the partial boundary, on which the superlinear radiation law is prescribed, is shrinking and is denoted as Γ1,t at time t, then the solution may exist globally as long as the surface area |Γ1,t| of Γ1,t decays fast enough. This paper asks the question that how fast should |Γ1,t| decay in order to have a bounded global solution? This question is of significant importance in realistic situations, such as the temperature control within a certain safe range. By taking advantage of the Neumann heat kernel, we conclude that a polynomial decay |Γ1,t| ∼ |Γ1|(1 + Ct) −β with any β > n − 1 suffices to ensure a bounded global solution.