Math 6410 • ODE

Wave Equation. Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with smooth boundary. Consider the initial boundary value problem \[ \begin {aligned} u_{tt} &= \Delta u,\quad x\in \Omega ,\quad t>0,\\ u(x,0) &= g(x),\quad x\in \Omega ,\\ u_t(x,0) &= 0,\quad x\in \Omega ,\\ u(x,t) &= 0,\quad x\in \partial \Omega ,\quad t>0. \end {aligned} \] Show that there exists a sequence of initial conditions \(\{g_j\}\subset C^2_0(\overline {\Omega })\) and a sequence of frequencies \(\{\omega _j\}\subset \mathbb {R}\) with \(\omega _j^2\to \infty \), such that each solution \(u^{(j)}(x,t)\) corresponding to initial condition \(g_j\) satisfies \[ u^{(j)}_{tt}(x,t)=-\omega _j^2u^{(j)}(x,t),\quad x\in \Omega ,\quad t>0. \] Moreover, prove that there is at most a countable set of \(\omega \) for which \(u_{tt}=-\omega ^2u\), no matter what initial conditions \(g\) are chosen.