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Stability in locally degenerate dual-phase-lag heat conduction. (English) Zbl 1455.35016
Summary: 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.
##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K51 Initial-boundary value problems for second-order parabolic systems 93D20 Asymptotic stability in control theory 35B35 Stability in context of PDEs
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