I have asked [this question on Mathematics Stack Exchange](https://math.stackexchange.com/questions/4818719/does-there-exist-research-about-equation-like-u-tt-detd2u
) yesterday, but there still is no reply. 

Does there exist research about equation like $$u_{tt}=\det(D_x^2 u)+\cdots\text{?}$$ That is to say, it contains second order time term $u_{tt}$ and the determination of Hessian of solution $\det(D_x^2 u)$ (we suppose that $u$ is convex, which means that the nonlinear operator is elliptic). Recently, I have read some papers concerning Hyperbolic mean curvature flow like the paper **The hyperbolic mean curvature flow** by K. Smoczyk and philippe G. LeFloch on JMPA. But does there exist some research like the above question, if yes, can you offer me some paper? Thank advance.