A multi-dimensional non-autonomous non-linear partial differential equation with senior partial derivative

We study the solutions of a multi-dimensional non-autonomous partial differential equation of arbitrary order which contains the senior partial derivative, an arbitrary nonlinearity with respect to an unknown function, and power nonlinearities with respect to its first derivatives. The separation of variables is applied for the investigation of this equation. We consider the cases when the right-hand side of the equation can be represented as a product of functions depending on some subsets of independent variables and, in particular, of functions of one variable. This equation is reduced either to ordinary differential equations or to partial differential equations of the lower dimension. We obtain particular solutions of the power, exponential, and logarithmic form, as well as a particular solution of the polynomial form, study the dependence of these solutions on the equation parameters, and find their existence conditions. We consider separately the case of a nonlinear non-autonomous equation of Bianci's type, the equation with only first derivatives with respect to every independent variable, and obtain the exact solutions to it.