Stability in locally degenerate dual-phase-lag heat conduction

Abstract

This work is devoted to the large time behaviour of a one-dimensional degenerate dual-phase-lag heat conduction system. Assume that those two delay parameters in the system are positive constants in a subregion of the spatial domain $[-1,1]$ and vanished in others. We obtain two stability results for this problem. First, we show that this system is exponentially stable under certain condition, which is consistent with the stability result for the system with delay parameters satisfying the same condition globally in the whole domain. Second, we show that the system is polynomially stable with decay rate $t^{-2}$ for the critical case, which is faster than the corresponding one with globally positive delay parameters, the decay rate of which is $t^{-1/2}$. The optimality of the decay rate $t^{-2}$ is further verified by the asymptotic spectral analysis for the system operator. Finally, some numerical experiments are presented to support these stability results.

COMMUNICATED BY:

• S. Hansen

Keywords:

• Dual-phase-lag heat conduction
• exponential stability
• polynomial stability
• spectrum

AMS subject classifications:

• 35B40
• 93D20

Acknowledgements

The authors would like to thank the editors and anonymous referees for their helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by the National Natural Science Foundation of China [grant number NSFC-61573252].