End effects in three-phase-lag heat conduction

Abstract

In this article we study the spatial behaviour of solutions for the three-phase-lag heat equation on a semi-infinite cylinder. First, we investigate the spatial evolution of solutions of an initial boundary-value problem with zero boundary conditions on the lateral surface of the cylinder. Under null initial data, a Phragmen–Lindelof alternative is obtained. An upper bound for the amplitude term in terms of the boundary data is also established. For the case of decay solutions, an improvement is obtained. We prove that the decay can be controlled by the exponential of a second-degree polynomial in the distance from the finite end of the cylinder. A class of non-standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T ₀ are assumed proportional to their initial values. These results are relevant in the context of the Saint-Venant Principle for heat conduction problems.

Keywords:

• three-phase-lag heat equation
• spatial evolution on a semi-infinite cylinder
• third derivative with respect to time

AMS Subject Classifications::

• 35L35
• 74F05
• 74G50

Acknowledgements

The research was supported by the Spanish Ministry of Science and Technology under project ‘Qualitative study of thermomechanical problems’ (MTM2006-03706). The author thanks to the anonymous referees for their useful comments.