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The paper considers an inverse problem and a direct problem for the Burgers equation in a domain with movable boundaries. With the help of an additional condition, a formula is obtained for determining the desired function from a direct problem for the loaded Burgers equation for the solvability of which we require a condition on the functions according to which the boundaries of the domain change. The solvability of direct problems is proved using a priori estimates and the methods of Faedo–Galerkin and functional analysis. [ABSTRACT FROM AUTHOR]

The work establishes the unique solvability of a boundary value problem for a 3D linearized system of Navier–Stokes equations in a degenerate domain represented by a cone. The domain degenerates at the vertex of the cone at the initial moment of time, and, as a consequence of this fact, there are no initial conditions in the problem under consideration. First, the unique solvability of the initial-boundary value problem for the 3D linearized Navier–Stokes equations system in a truncated cone is established. Then, the original problem for the cone is approximated by a countable family of initial-boundary value problems in domains represented by truncated cones, which are constructed in a specially chosen manner. In the limit, the truncated cones will tend toward the original cone. The Faedo–Galerkin method is used to prove the unique solvability of initial-boundary value problems in each of the truncated cones. By carrying out the passage to the limit, we obtain the main result regarding the solvability of the boundary value problem in a cone. [ABSTRACT FROM AUTHOR]

In the paper, we consider a coefficient inverse problem for the heat equation in a degenerating angular domain. It has been shown that the inverse problem for the homogeneous heat equation with homogeneous boundary conditions has a nontrivial solution up to a constant factor consistent with the integral condition. Moreover, the solution of the considered inverse problem is found in explicit form. In conclusion, statements of possible generalizations and the results of numerical calculations are given. [ABSTRACT FROM AUTHOR]