Decay rates for elastic-thermoelastic star-shaped networks

Abstract

This work discusses the asymptotic behaviour of a transmission problem on star-shaped networks of interconnected elastic and thermoelastic rods. Elastic rods are undamped, of conservative nature, while the thermoelastic ones are damped by thermal effects. We analyse the overall decay rate depending of the number of purely elastic components entering on the system and the irrationality properties of its lengths. First, a sufficient and necessary condition for the strong stability of the thermoelastic-elastic network is given. Then, the uniform exponential decay rate is proved by frequency domain analysis techniques when only one purely elastic undamped rod is present. When the network involves more than one purely elastic undamped rod the lack of exponential decay is proved and nearly sharp polynomial decay rates are deduced under suitable irrationality conditions on the lengths of the rods, based on Diophantine approximation arguments. More general slow decay rates are also derived. Finally, we present some numerical simulations supporting the analytical results.