This problem is from nonlinear wave equation.
From the following identity $$\partial_t\left(\displaystyle \frac{(\partial_t u)^2+|\nabla u|^2}{2}+\frac{u^6}{6}\right)-\operatorname{div}(\partial_t u\nabla u)=0$$
By multiplying by $\partial_{x_{i}} u$, show that $$\partial_t(\partial_t u\partial_{x_{i}}u)-\partial_{x_{i}}\left(\displaystyle \frac{(\partial_t u)^2-|\nabla u|^2}{2}-\frac{u^6}{6}\right)-\operatorname{div}(\partial_{x_{i}} u\nabla u)=0$$ where $i=1,2,3$
I have proved $$\displaystyle \partial_t\left(\frac{u^6}{6}\right)\partial_{x_{i}} u=(\partial_tu) \partial_{x_{i}} \left(\frac{u^6}{6}\right)$$ but I do not know what to do next.
Observe we have \begin{align} -u_{x_i}\partial_{tt} u + u_{x_i} \Delta u = u_{x_i} u^5 \end{align} then by product rule it follows \begin{align} & \partial_t (u_{x_i}\partial_tu)-\partial_t u_{x_i} \partial_t u -u_{x_i} \Delta u = -u_{x_i}u^5\\ \implies& \partial_t (u_{x_i}\partial_tu) - \frac{1}{2}\partial_{x_i}(\partial_t u)^2 -\operatorname{div}(u_{x_i}\nabla u)+\nabla \partial_{x_i}u\cdot \nabla u = -u_{x_i} u^5\\ \implies&\ \partial_t (u_{x_i}\partial_tu) - \frac{1}{2}\partial_{x_i}((\partial_t u)^2-|\nabla u|^2) -\operatorname{div}(u_{x_i}\nabla u)= -\partial_{x_i} \left(\frac{u^6}{6}\right) \end{align} Then we have the desired identity.