3D wave equations in sphere-symmetric domain with periodically oscillating boundaries (Q1864106)

The author deals with a linear nonhomogeneous wave equation of the form: \[ \begin{cases} \partial^2_t u-\Delta u=h(r,t) \quad & (r,t)\in \Omega,\\ u(r,t)=b_i(t) \quad & (r,t)\in \partial\Omega_i,\;i=1,2,\\ u(r,0)=\varphi (r),\;\partial_tu(r,0) =\psi(r), \quad & r\in[a_1(0), a_2(0)],\end{cases} \tag{1} \] where \(u\) is the unknown function of \((r,t)\), \(h(r,t)\) is a nonhomogeneous term and \(b_i(t)\), \(i=1,2\) are given boundary functions, \(\varphi(r)\) and \(\psi(r)\) are given initial data. Here \(\Omega\) is a time-periodic noncylindrical sphere-symmetric domain \(a_1(t)\leq r\leq a_2(t)\), \(t\in\mathbb{R}^1\) in \((x,y,z,t)\)-space with \(r=(x^2+y^2 +z^2)^{1/2}\). The inner and outer boundary of \(\Omega\) are denoted by \(\partial\Omega_1\) and \(\partial\Omega_2\) respectively. Under some natural conditions on \(h,b_i, \varphi,\psi\) and \(a_i\), the author shows that every solution of (1) is quasiperiodic in \(t\).