Multi-dimensional non-autonomous evolutionary equation of Monge–Ampère type

A multi-dimensional non-autonomous evolutionary equation of Monge–Ampère type is investigated. The left side of the equation contains the first time derivative with the coefficient depending on time, spatial variables and unknown function. The right side of the equation contains the determinant of Hessian matrix. The solutions with additive and multiplicative separation of variables are found. It is shown that a sufficient condition for the existence of such solutions is the representability of the coefficient of the time derivative as a product of functions in time and spatial variables. In the case when the time derivative coefficient is a function inverse to linear combination of spatial variables with coefficients depending on time, the solutions in the form of the quadratic polynomials in spatial variables is also found. The set of solutions in the form of the linear combination of functions of spatial variables with coefficients depending on time is obtained. Some reductions of the given equation to ordinary differential equations (ODE) in the cases when unknown function depends on sum of functions of spatial variables (in particular, sum of their squares) and function of the time are considered; in this case the functional separation of variables is used. Some reductions of the given equation to PDE of lower dimension are also found. In particular, the solutions in the form of function of the time and sum of squares of spatial variables as well as the solutions in the form of sum of such functions are obtained.