Existence Energy of Wave Equation

Question

I was just going trhough some properties of the wave equation, including the energy of the wave equation given by $E(t)=\int_{-\infty}^{\infty}u_t^2+c^2u_x^2 dx$, i.e the sum of kinetic and potential energy.

I never found something concerning the existence (convergence) of this integral, so my question is, why does this integral even exists?

Answer 1

The energy does not need to be finite. However, if you choose an initial condition $u(t=0)$ with $$E(t=0) =\int_{-\infty}^\infty[u_t^2(x,0) + c^2 u_x^2(x,0)] \,dx \leq \infty$$ and additionally $\lim_{|x|\to\infty} u_x u_t =0$ for all times, then the energy remains finite because $$\begin{align}\frac{d}{dt} E(t) &= \frac{d}{dt} \int_{-\infty}^\infty(u_t^2 + c^2 u_x^2) \,dx =2 \int_{-\infty}^\infty(u_t u_{tt} + c^2 u_x u_{xt} ) \,dx\\ &= 2c^2u_{x}u_t\Big|_{x=-\infty}^{\infty}-2 \int_{-\infty}^\infty \underbrace{(u_t u_{tt} - c^2 u_{xx}u_{t} )}_{u_t (u_{tt} -c^2 u_{xx}) =0}dx=0 \end{align}$$