Spatial behavior for solutions in heat conduction with two delays

Abstract

In this note, we investigate the spatial behavior of the solutions of the equation proposed to describe a theory for the heat conduction with two delay terms. We obtain an alternative of the Phragmén-Lindelöf type, which means that the solutions either decay or blow-up at infinity, both options in an exponential way. We also describe how to obtain an upper bound for the amplitude term. This is the first contribution on spatial behavior for partial differential equations involving two delay terms. We use energy arguments. The main point of the contribution is the use of an exponentially weighted energy function.

Keywords:

• three dual-phase-lag heat conduction
• two-temperature heat conduction
• spatial estimate
• energy argument
• equation with delay

AMS Subject Classifications:

• 35Q79
• 35E99
• 80A99

Acknowledgements

The authors thank to the anonymous referees their useful comments. This work is supported by the project “Análisis Matemático de las Ecuaciones en Derivadas Parciales de la Termomecánica” (MTM2013-42004-P) of the Spanish Ministry of Economy and Competitiveness.