Multidimensional Nonautonomous Evolution Monge–Ampère Type Equations

We study multidimensionalnonautonomous evolution Monge–Ampère type equations.The left-hand side of such equation contains the first time derivative with the coefficientdepending on time, space variables, and an unknown function. The right-hand side of the equationis the determinant of a Hessian matrix.We find the solutions by additive and multiplicativeseparation of variables and show thatthe representability of the coefficient of the timederivative as the product of functions of time and space variablesis a sufficient condition for the existence of such solutions.In the case that the coefficientof the time derivative is the inverse function to a linear combination of spacevariables with coefficients depending on time, we also give solutions in the form of the quadraticpolynomials in space variables. Also, we obtain the solution set in the form of thelinear combination of functions of space variables with time depending coefficients.We consider some reductions of the equation to ODEsin the cases that the unknown function depends on the sum of functions of space variables(in particular, the sum of their squares) and a function of the time; in thiscase we use the functional separation of variables. Some reductions are also foundof the given equation to PDEs of lower dimension. In particular, we find the solutions in the form offunction of the time and the sum of squares of space variables as well as the solutions inthe form of the sum of such functions.