Multi-dimensional second-order differentialequations with quadratic form on the first derivatives

We study multi-dimensional partial differential equations of the second order with quadratic form on the first derivatives, while considering the cases of autonomous and nonautonomous equations in which the coe cients of the quadratic form are functions of independent variables. The equation is reduced to the ordinary differential equation. We obtain solutions of the traveling wave type, self-similar solutions, and solutions in the form of quadratic polynomial and generalized polynomial. The existence conditions for these solutions are given. We prove that solutions of the traveling wave type exist for nonautonomous equations which are not invariant with respect to shift transformations of independent variables. A theorem giving the conditions of additive and multiplicative separation of variables is proved in the case when the matrice of coe cients of the quadratic form on the first derivatives is block-diagonal, and solutions of the aggregated traveling wave type are obtained in this case. These solutions are generalizations of some known solutions of the traveling wave type and depend on linear combinations of subsets of the set of independent variables. We study the dependence of the obtained solutions on the equation coe cients.