Blow-up problems for the heat equation with a local nonlinear Neumann boundary condition

This paper estimates the blow-up time for the heat equation ut=Δuut=Δu with a local nonlinear Neumann boundary condition: The normal derivative ∂u/∂n=uq∂u/∂n=uq on Γ1Γ1, one piece of the boundary, while on the rest part of the boundary, ∂u/∂n=0∂u/∂n=0. The motivation of the study is the partial damage to the insulation on the surface of space shuttles caused by high speed flying subjects. We show the finite time blow-up of the solution and estimate both upper and lower bounds of the blow-up time in terms of the area of Γ1Γ1. In many other work, they need the convexity of the domain Ω and only consider the problem with Γ1=∂ΩΓ1=∂Ω. In this paper, we remove the convexity condition and only require ∂Ω to be C2C2. In addition, we deal with the local nonlinearity, namely Γ1Γ1 can be just part of ∂Ω.