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Quenching for axisymmetric hypersurfaces under forced mean curvature flows

Hiroyoshi Mitake, Yusuke Oka, Hung Vinh Tran

Abstract

Here, we study the motion of axisymmetric hypersurfaces $\{Γ_t\}_{t\ge0}$ evolved by forced mean curvature flows in the periodic setting. We establish conditions that quenching occurs or does not occur in terms of the initial data and forcing term. We also study the locations where the quenching happens in some special cases.

Quenching for axisymmetric hypersurfaces under forced mean curvature flows

Abstract

Here, we study the motion of axisymmetric hypersurfaces evolved by forced mean curvature flows in the periodic setting. We establish conditions that quenching occurs or does not occur in terms of the initial data and forcing term. We also study the locations where the quenching happens in some special cases.
Paper Structure (4 sections, 11 theorems, 79 equations)

This paper contains 4 sections, 11 theorems, 79 equations.

Table of Contents

  1. Introduction and Main Theorems
  2. Quenching and Non-quenching
  3. $\varepsilon$-limit quenching
  4. The locations of quenching

Key Result

Proposition 1.1

Assume (A1). Let $\varepsilon, \alpha>0$, and $u^\varepsilon$ be the solution to eq:periodic, and let us denote $T_\varepsilon^\ast$ be the quenching time for eq:periodic. There exists $\varepsilon_0>0$ such that for all $\varepsilon\in(0,\varepsilon_0)$ if $\alpha>1$, then $T_\varepsilon^\ast<\inft

Theorems & Definitions (23)

  • Proposition 1.1
  • Theorem 1.1
  • Definition 1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1
  • proof : Proof of Proposition \ref{['prop:1']}
  • proof : Proof of Theorem \ref{['thm:1']}
  • Proposition 3.1
  • proof
  • ...and 13 more