### Abstract

We obtain a closed form analytic solution for the Dual Phase Lagging equation. This equation is a linear, time-independent partial differential equation modeling the heat distribution in a thin film. The spatial domain is of micrometer and nanometer geometries. We show that the solution is described by a semigroup, and obtain a basis of eigenfunctions. The closure of the set of eigenvalues contains an interval, and so the theory on Riesz spectral operator of Curtain and Zwart cannot be applied directly. The exponential stability and the approximate controllability is shown.

Original language | Undefined |
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Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 21 |

Publication status | Published - Feb 2010 |

### Publication series

Name | Memorandum / Department of Applied Mathematics |
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Publisher | Department of Applied Mathematics, University of Twente |

No. | 1917 |

ISSN (Print) | 1874-4850 |

ISSN (Electronic) | 1874-4850 |

### Keywords

- IR-69930
- METIS-277397
- MSC-35Q80
- MSC-35Q93
- MSC-35P10
- MSC-93B05
- MSC-74K35
- EWI-17434
- Thin film
- Semigroup
- Dual phase lagging

## Cite this

Heidari, H., Zwart, H. J., & Malek, A. (2010).

*Controllability and stability of 3D heat conduction equation in a submicroscale thin film*. (Memorandum / Department of Applied Mathematics; No. 1917). Enschede: University of Twente, Department of Applied Mathematics.