Optimal Energy Decay in a One-Dimensional Wave-Heat-Wave System

Abstract

Harnessing the abstract power of the celebrated result due to Borichev and Tomilov (Math. Ann. 347:455–478, 2010, no. 2), we study the energy decay in a one-dimensional coupled wave-heat-wave system. We obtain a sharp estimate for the rate of energy decay of classical solutions by first proving a growth bound for the resolvent of the semigroup generator and then applying the asymptotic theory of $C_0$-semigroups. The present article can be naturally thought of as an extension of a recent paper by Batty, Paunonen, and Seifert (J. Evol. Equ. 16:649–664, 2016) which studied a similar wave-heat system via the same theoretical framework.

Appendix—Entries for the Cofactor Matrix C in Sect. 3

$$\begin{aligned} c_{11}&= -\lambda ^{3/2}[e^{\sqrt{\lambda }}T_+(\lambda ) + e^{-\sqrt{\lambda }}T_-(\lambda )], \\ c_{12}&= \lambda ^2\cosh (\lambda )(e^{\sqrt{\lambda }}T_+ - e^{-\sqrt{\lambda }}T_-), \\ c_{13}&= -\lambda ^2\cosh (\lambda )[e^{\sqrt{\lambda }}T_+(\lambda ) +e^{-\sqrt{\lambda }}T_-(\lambda )], \\ c_{14}&= \lambda ^{3/2} \cosh (\lambda ), \\ c_{21}&= -\lambda [e^{\sqrt{\lambda }}T_+(\lambda ) - e^{-\sqrt{\lambda }}T_-(\lambda )],\\ c_{22}&= -\lambda ^2\sinh (\lambda )[e^{\sqrt{\lambda }}T_+(\lambda ) -e^{-\sqrt{\lambda }}T_-(\lambda )],\\ c_{23}&= \lambda ^2\sinh (\lambda )[e^{\sqrt{\lambda }}T_+(\lambda ) + e^{-\sqrt{\lambda }}T_-(\lambda )],\\ c_{24}&= -\lambda ^{3/2}\sinh (\lambda ), \\ c_{31}&= \lambda ^{3/2}\cosh (\lambda ), \\ c_{32}&= \lambda ^{5/2}\sinh (\lambda )\cosh (\lambda ), \\ c_{33}&= \lambda ^2\cosh ^2(\lambda ), \\ c_{34}&= -\lambda ^{3/2}[e^{\sqrt{\lambda }}T_+(\lambda ) e^{-\sqrt{\lambda }}T_-(\lambda )],\\ c_{41}&= -\lambda ^{3/2}\sinh (\lambda ), \\ c_{42}&= -\lambda ^{5/2} \sinh ^2(\lambda ), \\ c_{43}&= -\lambda ^2\cosh (\lambda )\sinh (\lambda ), \\ c_{44}&= -\lambda [e^{\sqrt{\lambda }}T_+(\lambda ) - e^{-\sqrt{\lambda }}T_-(\lambda )]. \end{aligned}$$