It's been some hours now that I am trying to find the Lagrangian density of the following hyperbolic PDE with variable coefficients and $c_{ij}=c_{ji}(x)$ $$ \partial^2_t u - c_{ij}\partial_i\partial_ju = 0 \quad \text{in} \quad \Omega \subset \mathbb{R}^n \times (0,\infty) \tag{$*$} \label{hypeq} $$ So, if we define, for example, the Lagrangian density to be $$ L(\nabla_{t,x}u,u,x,t) = \frac{1}{2}\big( u_t^2 - c_{ij}\partial_iu\partial_ju\big) $$ taking the first variation of the functional $$ I[u] = \int_{\mathbb{R}_+}\int_\Omega L(\nabla_{t,x}u,u,x,t)\ \mathrm{d}^nx \mathrm{d} t $$ we see that the associated Euler-Lagrange equation is (assuming I am correct) $$ \partial^2_tu - c_{ij}\partial_j\partial_iu - \color{red}{\partial_ic_{ij}\partial_ju} = 0 $$ Any ideas/hints on how to drop the second term, i.e. find the $L(\cdot)$ associated with \eqref{hypeq}?
Also, is there any possibility that a Lagrangian density does not exist at all for the above equation?