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The enclosure method for the heat equation using time-reversal invariance for a wave equation

  • Masaru Ikehata ORCID logo EMAIL logo
Veröffentlicht/Copyright: 8. November 2019

Abstract

The heat equation does not have time-reversal invariance. However, using a solution of an associated wave equation which has time-reversal invariance, one can establish an explicit extraction formula of the minimum sphere that is centered at an arbitrary given point and encloses an unknown cavity inside a heat conductive body. The data employed in the formula consist of a special heat flux depending on a large parameter prescribed on the surface of the body over an arbitrary fixed finite time interval and the corresponding temperature field. The heat flux never blows up as the parameter tends to infinity. This is different from a previous formula for the heat equation which also yields the minimum sphere. In this sense, the prescribed heat flux is moderate.

MSC 2010: 35R30; 35L05; 35K05

Award Identifier / Grant number: 17K05331

Funding statement: The author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 17K05331) of Japan Society for the Promotion of Science.

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Received: 2018-11-01
Revised: 2019-10-02
Accepted: 2019-10-15
Published Online: 2019-11-08
Published in Print: 2020-02-01

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