I have asked this question on math stack exchange yesterday, but there is no reply. See https://math.stackexchange.com/questions/4818719/does-there-exist-research-about-equation-like-u-tt-detd2u

Does there exist research about equation like $$u_{tt}=\det(D_{x}^{2}u)+\dots?$$ 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 this is elliptic). Recently, I have read some papers concerning Hyperbolic meaning curvature flow like the paperThe 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.