The first mixed problem for the nonstationary Lamé system (Q1697464)

The authors study the initial-boundary value problem for the nonstationary Lamé system in elastodynamic \[ \begin{aligned} &\rho u_{tt}=-\mu \Delta u+(\lambda+\mu)\nabla \text{div}\,u+f,\;(x,t)\in \mathcal{X}\times (0,T),\\ & u(x,0)=u_0(x),\;u_t(x,0)=u_1(x),\;x\in \mathcal{X},\\& u(x,t)=0,\;(x,t)\in \partial\mathcal{X}\times (0,T), \end{aligned} \] where \(\mathcal{X}\subset \mathbb{R}^3 \) is a bounded domain with smooth boundary and \(u:\mathcal{X}\times (0,T)\to \mathbf{R}^3\) is a displacement vector. An existence and uniqueness result for weak solutions is stated using Fourier series and the Galerkin method applied also to classical hyperbolic problems. The regularity of the solution is verified too.