CONDITIONS FOR THE EXISTENCE OF A UNIQUE SOLUTION TO THE BOUNDARY VALUE PROBLEM FOR A PARTIAL DERIVATIVE DIFFERENTIAL EQUATION WITH A NEGATIVE SIGN FOR THE HIGH DERATIVE

Abstract The article examines about the existence of a unique solution to a boundary value problem for a second-order partial differential equation with a negative sign at the highest derivative. This equation was obtained as a result of mathematical modeling of creeping movements in a two-layer viscous fluid, when the density of the lower layer is less than the density of the upper layer, and is limited on both sides by «walls». It is known that due to the difference in the layers, a violation of the equilibrium state of the boundary between them occurs; the materials in the bottom layer go up and the definition of the law of changes in the boundary between layers. Here the need arose to solve a differential equation, the form of which differs from the usual parabolic type equation in that the sign of the highest derivative is negative. This difference from a well-studies equation of parabolic type leads to the need to study the existence of a unique solution to the problem for such a partial differential equation.