Decay estimates for a thermoelastic system in waveguides
The authors investigate the long time behaviour of solutions to the initial-boundary value problem of linear thermoelastic equations in \(\{(t,x): t>0\), \(x\in \Omega\}\) with \(\Omega=(0,1)\times \mathbb{R}^{n-1}\) being a so-called waveguide: \[ \begin{aligned} &u_{tt}-\mu\Delta u -(\mu+\lambda): \operatorname{grad\,div}u+\gamma_1:\operatorname{grad}\theta=f(t,x),\\ &\theta_t-\kappa\Delta \theta+\gamma_2: \operatorname{div}u_t=g(t,x),\quad (t,x)\in \mathbb{R}_+\times \Omega,\\ &(u,u_t,\theta)(0,x)=(u^0,u^1,\theta^0),\quad x\in \Omega,\\ &u_1(t,x)=\partial_\nu u_2(t,x)=\cdots=\partial_\nu u_n(t,x)=\partial_\nu \theta(t,x)=0,\quad (t,x)\in \mathbb{R}_+\times \Omega. \end{aligned} \] They prove decay estimates of the solutions which play a key role in showing the global existence of solutions.
scientific article; zbMATH DE number 5692557
scientific article