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 u(t) = Delta(u) with a local nonlinear Neumann boundary condition: The normal derivative partial derivative u/partial derivative n = u(q) on Gamma(1), one piece of the boundary, while on the rest part of the boundary, partial derivative u/partial derivative 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 Gamma(1). In many other work, they need the convexity of the domain Omega and only consider the problem with Gamma(1) = partial derivative Omega. In this paper, we remove the convexity condition and only require partial derivative Omega to be C-2. In addition, we deal with the local nonlinearity, namely Gamma(1) can be just part of partial derivative Omega. (C) 2016 Elsevier Inc. All rights reserved.