Decay rates for 1-d heat-wave planar networks

Han ZJ, Zuazua E (2016)

Publication Type: Journal article

Publication year: 2016

Journal

Networks and Heterogeneous Media American Institute of Mathematical Sciences (AIMS)

Book Volume: 11

Pages Range: 655-692

Journal Issue: 4

DOI: 10.3934/nhm.2016013

Abstract

The large time decay rates of a transmission problem coupling heat and wave equations on a planar network is discussed. When all edges evolve according to the heat equation, the uniform exponential decay holds. By the contrary, we show the lack of uniform stability, based on a Geometric Optics high frequency asymptotic expansion, whenever the network involves at least one wave equation. The (slow) decay rate of this system is further discussed for star-shaped networks. When only one wave equation is present in the network, by the frequency domain approach together with multipliers, we derive a sharp polynomial decay rate. When the network involves more than one wave equation, a weakened observability estimate is obtained, based on which, polynomial and logarithmic decay rates are deduced for smooth initial conditions under certain irrationality conditions on the lengths of the strings entering in the network. These decay rates are intrinsically determined by the wave equations entering in the system and are independent on the heat equations.

Authors with CRIS profile

Enrique Zuazua Iriondo

Involved external institutions

Tianjin University (TJU) / 天津大学 CN China (CN) University of Deusto / Universidad de Deusto ES Spain (ES)

How to cite

APA:

Han, Z.-J., & Zuazua, E. (2016). Decay rates for 1-d heat-wave planar networks. Networks and Heterogeneous Media, 11(4), 655-692. https://dx.doi.org/10.3934/nhm.2016013

MLA:

Han, Zhong-Jie, and Enrique Zuazua. "Decay rates for 1-d heat-wave planar networks." Networks and Heterogeneous Media 11.4 (2016): 655-692.

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