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?
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 $$\begin{align}\frac{d}{dt} E(t) &= \frac{d}{dt} \int_{-\infty}^\infty(u_t^2 + c^2 u_x^2) \,dx =2 \int_{-\infty}^\infty(u_t u_{tt} + c^2 u_x u_{xt} ) \,dx\\ &= 2c^2u_{x}u_t\Big|_{x=-\infty}^{\infty}-2 \int_{-\infty}^\infty \underbrace{(u_t u_{tt} - c^2 u_{xx}u_{t} )}_{u_t (u_{tt} -c^2 u_{xx}) =0}dx=0 \end{align}$$