On multiplicative multi-dimensional partial differential equations

The class of multiplicative partial differential equations is considered. The left side of the equation is represented in the form of the product of linear differential expressions of arbitrary order. The right side of the equation is a function of independent variables and unknown function. The solutions with additive, multiplicative and combined separation of variables are received for the equation with one-dimensional linear operators and factorized right side. The initial equation is reduced either to ordinary differential equation or to partial differential equation of lower dimension. It is shown that if the operators in the left side are homogeneous then the equation has the solutions in the form of generalized polynomials. Also the self-similar solutions are founded and the sufficient conditions of their existence are formulated for the equation with homogeneous operators. The traveling wave type solutions are received for the case of operators with constant coefficients and the right side depending on the linear combination of independent variables. It is shown that if the right side of equation depends on several linear combinations on subsets of independent variables, then there are multi-dimensional traveling wave type solutions depending on these linear combinations. A solution is obtained in the form of an expansion with respect to eigenfunctions of the kernels of linear operators in the left side of equation. Also some equations with multi-dimensional linear operators are investigated. In particular, the solutions with separation of variables for the case of factorized multi-dimensional operators, the of multi-dimensional traveling wave type solutions, and the solutions in the form of functions on arguments of more complicated structure are obtained. The cases when the right side of equation contains power and exponential nonlinearities on unknown function are considered.