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I am solving a initial value problem for the wave equation $$ u_{tt}=u_{xx}\ \ in \ \ \mathbb{R}\times (0,\infty), \ \ \ u=g, \ u_{t}=h \ \ on \ \ \mathbb{R}\times \{0\} $$ for some com[actly suppoerted functions $f,g\in C_{c}(\mathbb{R})$. Let $u$ be a solution to the wave equation above. The kinetic energy $k(t)$ and the potential energy $p(t)$ of $u$ are defined respectively $$ k(t)=\int_{\mathbb{R}}u_t^2(x,t)dx, \ \ \ \ p(t)=\int_{\mathbb{R}}u_x^2(x,t)dx. $$ I have trouble in showing that $k(t)=p(t)$ for sufficiently large $t$. I would appreciate it if someone could help me proving this equality for large time $t$.

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