D HEAT-WAVE PLANAR NETWORKS Zhong-Jie Han Enrique Zuazua
NETWORKS AND HETEROGENEOUS MEDIA doi:10.3934/nhm.2016013 c American Institute of Mathematical Sciences Volume 11, Number 4, December 2016 pp. 655{692 DECAY RATES FOR 1 − d HEAT-WAVE PLANAR NETWORKS Zhong-Jie Han Department of Mathematics Tianjin University Tianjin 300072, China Enrique Zuazua (1) Deusto Tech, University of Deusto 48007 Bilbao, Basque Country - Spain (2) Departamento de Matem´aticas Universidad Aut´onomade Madrid 28049 Madrid, Spain (Communicated by Benedetto Piccoli) 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 expo- nential 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 poly- nomial 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. 1. Introduction. In recent years, hyperbolic-parabolic coupled models have been studied extensively due to their applications in analyzing fluid-structure interac- tions, which are crucial in many scientific and engineering areas, such as airflow along the aircraft, deformation of heart valves, the process of mixing and so on (see 2010 Mathematics Subject Classification.
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