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

Abstract

This paper estimates the blow-up time for the heat equation u t = u with a local nonlinear Neumann boundary condition: The normal derivative ∂u/∂n = u q on 1 , 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 1 . In many other work, they need the convexity of the domain and only consider the problem with 1 = ∂ . In this paper, we remove the convexity condition and only require ∂ to be C 2 . In addition, we deal with the local nonlinearity, namely 1 can be just part of ∂ .

Lifespan estimates via Neumann heat kernel

“…In fact, if the solution u to (1.1) is smooth, then it is straightforward to obtain (2.12) based on the properties of N (x, y, t) in Corollary 2.4. Now although u is not smooth near the boundary ∂Ω or near the initial time t = 0, by taking advantage of Lemma 2.5, we are able to verify (2.12) in a way similar to the proof for (1.12) in [35]. In contrast to (1.12), the formula (2.12) does not contain any term that may cause the jump relation along the boundary ∂Ω.…”

“…One should also note that C and C i may represent different constants in different places. The recent paper [35] studied (1.1) systematically and the motivation was the disaster of the Space Shuttle Columbia (see Figure 2) in 2003, we refer the reader to that paper for the detailed discussion of the background. As a summary of its conclusions, [35] first established the local existence and uniqueness theory for (1.1) in the following sense: there exist T > 0 and a unique solution u in C 2,1 Ω × (0, T ]…”

“…Moreover, it is shown that this unique solution u is strictly positive when t > 0. We want to remark here that the solution constructed in [35] through the heat potential technique automatically satisfies (1.7) due to u: temperature Γ 1 Ω Γ 2 Figure 2: Space Shuttle Columbia a generalized jump relation (See Theorem A.3 in [35]). The purpose of imposing this additional restriction (1.7) to the local solution is to ensure the uniqueness through the Hopf's lemma, it is not clear whether the uniqueness will still hold without this restriction.…”

“…Meanwhile, [35] also provides a lower bound for T * . Later in [36], it improves the lower bound as below.…”

“…The existence of the coefficients 2's in (1.11) is due to the jump relation of the single-layer heat potential when x ∈ ∂Ω (see e.g. Corollary Appendix A.2 in [35] or Theorem 9.5, Sec. 2, Chap.…”