Unique solvability of second order nonlinear totally characteristic equations

. We consider a second order singular nonlinear partial differential equation of the form ( t ∂ t ) 2 u = F ( t , x , u , ∂ x u , ∂ 2 x u , t ∂ t u , t ∂ t ∂ x u ) , where F is assumed to be continuous in t and holo-morphic with respect to the other variables. Under certain conditions, we prove that the equation has a unique solution that is continuous in t and holomorphic in x .