Multidimensional thermoelasticity for nonsimple materials – well-posedness and long-time behavior

Abstract

An initial-boundary value problem for the multidimensional type III thermoelaticity for a nonsimple material with a center of symmetry is considered. In the linear case, the well-posedness with and without a (second-order in space) Kelvin–Voigt and/or frictional damping in the elastic part as well as the lack of exponential stability in the elastically undamped case are proved. Further, a frictional damping for the elastic component is shown to lead to exponential stability. A Cattaneo-type hyperbolic relaxation for the thermal part is introduced and the well-posedness and uniform stability under a nonlinear frictional damping are obtained using a compactness-uniqueness-type argument. Additionally, a connection between exponential stability and exact observability of unitary strongly continuous groups is established.

Keywords:

• Thermoelasticity
• nonsimple materials
• well-posedness
• uniform stability
• hyperbolic relaxation

AMS Subject Classifications:

• 35B40
• 35G61
• 35Q74
• 74F05
• 93D20

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work has been partially funded by the Young Scholar Fund at the University of Konstanz, Germany. MP has been supported by the ERC-CZ Project LL1202 ‘MOdelling REvisited + MOdel REduction’ at Charles University in Prague, Czech Republic and the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173 at Karlsruhe Institute of Technology, Germany.