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.
Citation: Zhong-Jie Han, Enrique Zuazua. Decay rates for heat-wave planar networks[J]. Networks and Heterogeneous Media, 2016, 11(4): 655-692. doi: 10.3934/nhm.2016013
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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.
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