EXACT SOLUTIONS OF NON-CLASSICAL NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

Since the middle of the 20th century, Sobolev-type equations containing a superposition of differentiation with respect to time and the Laplacian have been actively studied. Such equations can be used in hydrodynamics, semiconductor physics and other areas of physics. There are extensive studies on the qualitative theory of Sobolev equations: on issues of unique solvability, estimates of the lifetime of solutions, asymptotic behavior of solutions. On the other hand, such equations are rare in the literature on exact solutions. Exact solutions can be useful for explaining of some physical effects, for better understanding the qualitative properties of the equation and for testing and improving numerical methods. The report is devoted to the construction of exact solutions of several nonlinear Sobolev-type equations. The work used the construction of solutions of a special type, Painlevé analysis and group analysis.