Complex valued Hamilton Jacobi equation

Question

Let $g_{ij}(t,x)$ be a metric tensor with dependence on t,x. Consider $$\partial_t u(t,x) = i\sqrt{\sum_{i,j} g_{ij}\partial_iu\partial_ju},u(0,x)=u_0(x).$$ Where $u(t,x):\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{C}$ is complex valued. Assume $u_0$ is real valued. It seems the usual method of Hamilton Jacobi Equation does not apply here. I want to know if there's any local existence results and how to solve it if possible. Thanks for any potential help and references!