Equations and systems of elliptic type with power-law nonlinearities are considered. Such equations are found in modeling distributed robotic formations, as well as in chemical kinetics, biology, astrophysics, and many other fields. The problem of constructing multidimensional exact solutions is studied. It is proposed to use a special type of ansatz that reduces the problem to solving systems of algebraic equations. A number of multiparameter families of new exact multidimensional solutions (both radially symmetric and anisotropic) represented by explicit formulas are obtained. Examples are given to illustrate the exact solutions found.

We consider a multidimensional pantograph-type nonlinear diffusion equation with a linearly increasing time delay and scaling with respect to spatial variables in the source (sink). It is proposed to construct exact solutions by the reduction method using two ansatzes with a quadratic dependence on spatial variables. The dependence of the solution on spatial variables is found from a system of algebraic equations, and the dependence on time is found from a system of ordinary differential equations with a linearly increasing delay of the argument. A number of examples of exact solutions are given, both radially symmetric and anisotropic with respect to spatial variables.

The equations of motion of the Goryachev–Sretensky gyrostat are studied. All stationary solutions are found on the invariant set of the zero level of the area integral, and their stability is analyzed. For the case where the suspension point coincides with the center of mass and the action of a gyroscopic moment is of a special type, integration by quadratures is performed.

We consider a nonlinear second-order ordinary differential equation of a special formwhose particular case arises when constructing exact solutions of the nonlinear heat equation witha power-law coefficient. Conditions are obtained for the parameters under which the equationadmits a single integration. A number of examples of constructing exact solutions expressed interms of elementary functions or in terms of the Lambert function are given.

We consider the multidimensional nonlinear diffusion equation with a power coefficient. Using some multidimensional quadratic ansatz, we seek for generalized automodel solutions and find new exact solutions in elementary and special functions in case of various exponents. We distinguish the events that the solutions are radially symmetric or spatially anisotropic and exhibit a series of examples demonstrating the novelty of the solutions.

The paper studies the equations of motion of a gyrostat around a fixed point under the action of a moment of forces. A general form of the moment that endows the equations with the three first integrals (energy, area, and geometric integrals) is established. Conditions for the existence of an additional integral as well as for the existence of a particular integral analogous to the Hess integral are derived. Stationary motions of the gyrostat are identified, and their stability in the Lyapunov sense is analyzed using Chetaev’s method and the integrals obtained.

A nonlinear multidimensional heat equation with a power-law coefficient is studied. It isproposed to construct its exact solutions by the multidimensional reduction method based on theuse of a special ansatz. As a result of the reduction, the problem is reduced to solving systems ofmatrix-vector algebraic equations that determine the dependence on the spatial variables andintegrating ordinary differential equations that determine the dependence on time. For a numberof examples with various values of the exponents, explicit expressions are obtained in terms ofelementary functions for exact multidimensional solutions, including those anisotropic in thespatial variables. The exact solutions found can be useful when constructing approximatesolutions of boundary value problems for the nonlinear heat equation using numerical methodsleading to the need to solve high-dimensional systems of equations.

We study the system of two fourth-order nonlinear hyperbolic partial differentialequations. The right-hand sides of the equations contain double Laplace operators and the squaresof the gradients of the sought functions. Such equations, close to the Boussinesq equation and theNavier–Stokes equations, occur in problems of hydrodynamics. We propose to search for a solutionin the form of an ansatz containing quadratic dependence on the spatial variables and arbitraryfunctions of time. The use of the ansatz allows us to decompose the process of finding thecomponents of the solution depending on the space variables and time. For finding the dependenceon the space variables, it is necessary to solve an algebraic system of matrix, vector, and scalarequations. We find the general solution to this system in parametric form. In finding thetime-dependent components of the solution to the original system, there arises a system ofnonlinear ordinary differential equations. In the particular case when the squares of the gradientsare not included in the system, we establish the existence of exact solutions of a certain kind tothe original system expressed through arbitrary harmonic functions of the space variables andexponential functions of time. Some examples are given of the constructed exact solutionsincluding solutions periodic in time and anisotropic in space variables. The exact solutions can beused to verify numerical methods for the approximate construction of the solutions to appliedboundary value problems.

A system of nonlinear parabolic differential equations is being studied, considered as a distributed mathematical model of the process of examining three-dimensional space by interacting robots of two types. Parametric families of exact solutions have been built that can be used to form the control of the survey process by creating the necessary densities at the border of the area that is the base for robots.

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We consider a nonlinear kinetic model described by a system of two equations in partial derivatives of the elliptic type with exponential nonlinearities. We propose to construct exact solutions of the mathematical model in the class of logarithms from quadratic functions of spatial variables. The solution coefficients of the model are found from systems of square matrix and linear vector equations. In particular, the proposed approach is used to construct anisotropic solutions to the Liouville equation, often used as a mathematical model of stationary distributions in plasma physics. We illustrate the results by a number of examples.