Том 61, № 6 (2025)

We consider evolutionary multidimensional generalized Monge–Amp`ere equations, the right parts of which, in addition to the determinant of the Hesse matrix, may depend on the Laplace operator and the gradient of the desired function. A variant of the reduction method for constructing exact multidimensional solutions of the generalized Monge–Amp`ere evolution equations using separation of variables is proposed. Multivariate exact solutions expressed explicitly through elementary functions and through solutions of ordinary differential equations are obtained. A number of examples of exact solutions, both radially symmetric and anisotropic in spatial variables, expressed through combinations of elementary functions are given.

The first initial-boundary value problem for Petrovskii second-order parabolic system in a bounded domain on the plane is investigated. The coefficients of the system satisfy the double Dini condition. The lateral boundaries of the domain admit cusps at the initial moment of time. The question of the existence of a solution to that problem in the space of functions that are continuous and bounded together with their highest derivatives in the closure of the domain is studied. An integral representation of that solution is obtained. Corresponding estimates are established.