Asymptotic behaviour for wave equation with time-dependent coefficients (Q2465925)

The author considers the following strictly hyperbolic Cauchy problem: \[ (\partial^2_t- c^2(t)\Delta_x) u(x,t)= 0,\quad (x,t)\in\mathbb{R}^N\times (0,\infty),\tag{1} \] \[ u(x,0)= u_0(x),\quad \partial_t u(x,0)= u_1(x),\quad x\in\mathbb{R}^N.\tag{2} \] The author shows that if \(c(t)\) tends to \(C_\infty\) rapidly as \(t\) goes to infinity, then each solution of (1)--(2) is asymptotically free. While the convergence rate of \(c(t)- C_\infty\) is slow, the author proves that there exists a solution of (1)--(2) which is not asymptotically free.