Multi-dimensional non-autonomous evolutionary equation of Monge–Ampère type

I. V. Rakhmelevich

Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 1, pp. 64-80

Résumé

A multi-dimensional non-autonomous evolutionary equation of Monge–Ampère type is investigated. The left side of the equation contains the first time derivative with the coefficient depending on time, spatial variables and unknown function. The right side of the equation contains the determinant of Hessian matrix. The solutions with additive and multiplicative separation of variables are found. It is shown that a sufficient condition for the existence of such solutions is the representability of the coefficient of the time derivative as a product of functions in time and spatial variables. In the case when the time derivative coefficient is a function inverse to linear combination of spatial variables with coefficients depending on time, the solutions in the form of the quadratic polynomials in spatial variables is also found. The set of solutions in the form of the linear combination of functions of spatial variables with coefficients depending on time is obtained. Some reductions of the given equation to ordinary differential equations (ODE) in the cases when unknown function depends on sum of functions of spatial variables (in particular, sum of their squares) and function of the time are considered; in this case the functional separation of variables is used. Some reductions of the given equation to PDE of lower dimension are also found. In particular, the solutions in the form of function of the time and sum of squares of spatial variables as well as the solutions in the form of sum of such functions are obtained.

Bibliographie

[1] Polyanin, A. D. and Zaytsev, V. F., Handbook of Nonlinear Partial Differential Equations, 2nd Ed., Chapman, Boca Raton; Hall-CRC Press, London, 2012, 1841 pp. | MR

[2] Rozhdestvenskii, B. L. and Yanenko, N. N., Systems of Quasylinear Equations and Their Applications to Gas Dynamics, Nauka Publ, M., 1988 (in Russian) | MR

[3] Khabirov, S. V., “Nonisentropic One-Dimensional Gas Motions Constructed with by Means of Contact Group of the Nonhomogeneous Monge–Ampère Equation”, Mathematics of the USSR-Sbornik, 71:2 (1992), 447–462 | DOI | MR | Zbl | Zbl

[4] Shablovskii, O. N., “Parametric Solution for the Monge–Ampere Equation and Gas Flow with Variable Entropy”, Tomsk State University Journal of Mathematics and Mechanics, 2015, no. 1(33), 105–118 (in Russian) | DOI

[5] Pogorelov, A. V., Multi-Dimensional Monge–Ampère Equation, Nauka Publ, M., 1988 (in Russian) | MR

[6] Guberman, I. Ya., “On Existence of Many Solutions of Dirichlet Problem for Multi-Dimensional Equation of Monge–Ampère Type”, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1965, no. 4, 54–63 (in Russian) | MR | Zbl

[7] Rakhmelevich, I. V., “Multi-Dimensional Monge–Ampère Equation with Power-Law Non-Linearities on the First Derivatives”, Proceedings of Voronezh State University. Ser. Physics. Mathematics, 2020, no. 2, 86–98 | MR

[8] Ibragimov, N. H., Symmetries, Exact Solutions and Conservation Laws, CRC Handbook of Lie Groups to Differential Equations, 1, CRC Press, Boca Raton, 1994, 429 pp. | MR

[9] Rakhmelevich, I. V., “On the Solutions of Two-Dimensional Monge–Ampère Equation with Power-Law Non-Linearity on the First Derivatives”, Tomsk State University Journal of Mathematics and Mechanics, 2016, no. 4(42), 33–43 (in Russian) | DOI | MR

[10] Galaktionov, V. A. and Svirshchevskii, S. R., Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman and Hall/CRC Press, Boca Raton, 2006 | MR

[11] Gutierres, C. E., The Monge–Ampère Equation, Birkhauser, Boston, 2001 | MR

[12] Taylor, M. E., Partial Differential Equations III. Nonlinear Equations, Springer-Verlag, New York, 1996 | MR

[13] Polyanin, A. D. and Zhurov, A. I., Variables Separation Methods and Exact Solutions of Nonlinear Equations in Mathematical Physics, IPMech RAN Publ, M., 2020 (in Russian)

[14] Faddeev, D. K. and Sominsky, I. S., Collection of Problems on Highest Algebra, M., 1977 (in Russian) | MR