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Consider the one-dimensional wave equation on $R\times (0,\infty)$, $$u_{tt}-u_{xx}=0.$$ Prove that $$ u(x,t)=\left\{ \begin{array}{ll} 3,|x|<t\\ 0,|x|>t\\ \end{array} \right. $$

satisfies $\int_{0}^{\infty}\int_{R}u(\,f_{tt}-f_{xx})\,dx\,dt=0$, where $f$ is a test function ($\,f$ belongs to $C^{\infty}$ and has a compact support).

$\int_{0}^{\infty}\int_{R}u(f_{tt}-f_{xx})dxdt=0$ $\Leftrightarrow$ $\int_{0}^{\infty}\int_{-t}^{t}3(f_{tt}-f_{xx})dxdt=0$ $\Leftrightarrow$ $\int_{0}^{\infty}\int_{-t}^{t}f_{tt}\,dxdt=\int_{0}^{\infty}\int_{-t}^{t}f_{xx}\,dxdt$.

I don't know how to prove the equality of the last bit.

Yiorgos S. Smyrlis
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Jack
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1 Answers1

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Integrating by parts (given suitable initial conditions), we have a more general result:

$$ \int f(u_{tt}-u_{xx}) = - \int f_t u_t + \int f_xu_x = \int f_{tt} u - \int f_{xx}u = \int u (f_{tt}-f_{xx}). $$

Randall
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