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

• Published in: Journal of Differential Equations Vol. 261; no. 5; pp. 2738 - 2783
• Main Authors: Yang, Xin, Zhou, Zhengfang
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 05.09.2016
• Subjects: Blow-up time; Heat equation; Local nonlinear; Neumann boundary condition; 35K20; Blow-up time; 35K60; Heat equation; 35K08; Neumann boundary condition; 35K05; Local nonlinear
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2016.05.011

This paper estimates the blow-up time for the heat equation ut=Δu with a local nonlinear Neumann boundary condition: The normal derivative ∂u/∂n=uq on Γ1, one piece of the boundary, while on the rest part of the boundary, ∂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 Γ1. In many other work, they need the convexity of the domain Ω and only consider the problem with Γ1=∂Ω. In this paper, we remove the convexity condition and only require ∂Ω to be C2. In addition, we deal with the local nonlinearity, namely Γ1 can be just part of ∂Ω.