Large time behavior of temperature in two-phase heat conductors

TL;DR

This paper investigates the long-term behavior of temperature in a two-media heat conduction problem, showing conditions for stabilization and examples of oscillation, advancing understanding of heat diffusion in heterogeneous media.

Contribution

It provides new insights into the conditions under which temperature stabilizes or oscillates in two-phase heat conductors over time.

Findings

Temperature stabilizes to a constant under certain geometric conditions.

Temperature can oscillate and not stabilize in general cases.

The study offers examples illustrating both stabilization and oscillation behaviors.

Abstract

We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media with different constant conductivities. The large time behavior of temperature, the solution of the problem, is studied when initially temperature is assigned to be 0 on one medium and $1$ on the other. We show that under a certain geometric condition of the configuration of the media, temperature is stabilized to a constant as time tends to infinity. We also show by examples that temperature in general oscillates and is not stabilized.