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Lifespan estimates via Neumann heat kernel

  • Xin YangEmail author
  • Zhengfang Zhou
Article
  • 18 Downloads

Abstract

We study the lower bound of the lifespan \(T^{*}\) for the heat equation \(u_t=\Delta u\) in a bounded domain \(\Omega \subset \mathbb {R}^{n}(n\ge 2)\) with positive initial data \(u_{0}\) and a nonlinear radiation condition on partial boundary: the exterior normal derivative \(\partial u/\partial n=u^{q}\) on \(\Gamma _1\subseteq \partial \Omega \) for some \(q>1\), while \(\partial u/\partial n=0\) on the other part of the boundary. Previously, under the convexity assumption of \(\Omega \), the asymptotic behaviours of \(T^{*}\) on the maximum \(M_{0}\) of \(u_{0}\) and the surface area \(|\Gamma _{1}|\) of \(\Gamma _{1}\) were explored. This paper is intended to remove the convexity assumption of \(\Omega \) since it is very restrictive in real applications. We will show that as \(M_{0}\rightarrow 0^{+}\), \(T^{*}\) is at least of order \(M_{0}^{-(q-1)}\) which is optimal, and meanwhile prove that as \(|\Gamma _{1}|\rightarrow 0^{+}\), \(T^{*}\) is at least of order \(|\Gamma _{1}|^{-\frac{1}{n-1}}\) for \(n\ge 3\) and \(|\Gamma _{1}|^{-1}\big /\ln \left( |\Gamma _{1}|^{-1}\right) \) for \(n=2\). The order of \(T^{*}\) on \(|\Gamma _{1}|\) when \(n=2\) is almost optimal. Instead of the usual energy method, the proofs in this paper are carried out by carefully analysing the representation formula of u in terms of the Neumann heat kernel.

Keywords

Blow-up Lifespan estimate Lower bound Neumann heat kernel 

Mathematics Subject Classification

35K20 35B44 35K08 35C15 

Notes

Acknowledgements

The authors appreciate the referees for their careful reading and many helpful suggestions which make the paper more clearly.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of CincinatiCincinnatiUSA
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

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