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# Dissipative structures for thermoelastic plate equations in R<sup>n</sup>

Type of Publication: | Working Paper/Technical Report |

URI (citable link): | http://nbn-resolving.de/urn:nbn:de:bsz:352-0-300753 |

Author: | Racke, Reinhard; Ueda, Yoshihiro |

Year of publication: | 2015 |

Series: | Konstanzer Schriften in Mathematik ; 340 |

Summary: |
We consider the Cauchy problem in R^ n for linear thermoelastic plate
equations where heat conduction is modeled by either the Cattaneo law or by the Fourier law – described by the relaxation parameter τ , where τ > 0 corresponds to Cattaneo’s law and τ = 0 corresponds to Fourier’s law. Additionally, we take into account possible inertial eﬀects characterized by a parameter µ ≥ 0, where µ = 0 corresponds to the situation without inertial terms. For the Catteneo system without inertial term, being a coupling of a Schrödinger type equation (the elastic plate equation) with a hyperbolic system for the temperature and the heat ﬂux, we shall show that a regularity-loss phenomenon appears in the asymptotic behavior as time tends to inﬁnity, while this is not given in the standard model where the Cattaneo law is replaced by the standard Fourier law. This kind of eﬀect of changing the qualitative behavior when moving from Fourier to Cattaneo reﬂects the eﬀect known for bounded domains, where the system with Fourier law is exponentially stable while this property is lost when going to the Cattaneo law. In particular, we shall describe in detail the singular limit as τ → 0. For the system with inertial term we demonstrate that it is of standard type, not of regularity loss type. The corresponding limit of a vanishing inertial term is also described. All constants appearing in the main results are given explicitly, allowing for quantitative estimates. The optimality of the estimates is also proved. |

Subject (DDC): | 510 Mathematics |

Link to License: | Terms of use |

Bibliography of Konstanz: | Yes |

Checksum:
MD5:c40b26ffc594ac5f00670915ec418543

RACKE, Reinhard, Yoshihiro UEDA, 2015. Dissipative structures for thermoelastic plate equations in R^{n}

@techreport{Racke2015Dissi-31959,
series={Konstanzer Schriften in Mathematik},
title={Dissipative structures for thermoelastic plate equations in R^{n}},
year={2015},
number={340},
author={Racke, Reinhard and Ueda, Yoshihiro}
}

Racke-0-300753.pdf | 220 |