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

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.

The enclosure method for inverse obstacle scattering over a finite time interval: VI. Using shell-type initial data

“…of input and output data yields one information. This gives a complete answer to the question raised in [10].…”

“…In [10], to generate a suitable heat flux we solve the Cauchy problem for an auxiliary wave equation with a parameter.…”

“…Another proof is based on the representation of the Fourier transform of v(x, T ; τ ) together with ∂ t v(x, T ; τ ) with respect to x. See the proof of Proposition 3.1 in [10].…”

“…The heat equation is a typical and an important example. However, in [10] we introduced an auxiliary equation which is a wave equation with a large parameter and using the time-reversal invariance of this equation, we found an extraction formula of the minimum radius of the sphere centered at an arbitrary given point in the whole space and enclosing all the unknown cavities inside the body. Since the wave equation therein contains a large parameter, the input heat flux also depends on the same parameter.…”

“…The idea yields the same information as above by using a single set of input and output data on the surface of the body. To make the essential difference from the idea in [8,10] clear and show the applicability to various inverse obstacle problems, we consider two inverse obstacle problems governed by the heat equation and a coupled system of the elastic wave and heat equations appearing in the linear theory of thermoelasticity.…”