Math 6410 • ODE

Suppose \(u(x,t): x\in \mathbb {R}\), \(t\in [0,\infty )\) is of class \(C^\infty \) and that \(u\) is a solution of \[ \begin {cases} u_{tt}=u_{xx}\\ u(x,0)=g(x)\\ u_t(x,0)=h(x). \end {cases} \] Suppose \(g\) and \(h\) are in \(C_0^\infty \). (\(C^\infty \) functions with compact support.) Define \[ k(t)=\frac 12\int _{-\infty }^{\infty }u_t^2(x,t)\,dx\quad \text {(kinetic energy)} \] and \[ p(t)=\frac 12\int _{-\infty }^{\infty }u_x^2(x,t)\,dx\quad \text {(potential energy)} \] Prove that \(k(t)+p(t)\) is constant and that for sufficiently large times \(t\), \(k(t)=p(t)\). Hint: d’Alembert’s solution.