APPLICATION OF A NONLOCAL BOUNDARY VALUE PROBLEM IN PROCESS CONTROL IN GASEOUS AND LIQUID ENVIRONMENTS
Abstract
The paper presents the results of a study of the internal boundary value problem for a mixed hyperbolic-parabolic equation characterized by the presence of multiple characteristics. The domain in which the equation is considered is finite and simply connected, consisting of two parts: hyperbolic and parabolic. This division of the domain is determined by the specific nature of the equation under consideration. The boundary between the hyperbolic and parabolic regions is a degeneracy line, where the transition from a hyperbolic to a parabolic equation occurs. A key feature of the problem is the boundary condition specified in the hyperbolic part of the region. This condition relates the values of the desired function at points belonging to the characteristics of the region, which adds significant complexity to the problem. The purpose of this work is to prove the existence and uniqueness theorem for the given boundary value problem. The proof is conducted for various cases determined by the discriminant of the cubic characteristic equation associated with the partial differential equation. Different values of the discriminant indicate different qualitative properties of the solutions and require an individual approach to the proof. To prove the uniqueness of a solution, the energy integral method is used. This method allows one to establish certain energy inequalities that relate the solution to the given functions and their point values in the boundary condition. The resulting inequalities impose constraints on the permissible values of the problem parameters, guaranteeing the uniqueness of the solution. Violating these constraints may lead to the problem being ill-posed or to the absence of a unique solution. The question of the existence of a solution is investigated by reducing the problem to an equivalent system of functional relations linking the traces of the desired solution and its derivative on the degeneracy line. These traces are considered separately for the hyperbolic and parabolic parts of the domain. Further investigation reduces to solving a Fredholm integral equation of the second kind. It is important to note that the kernel of this integral equation has a weak singularity, while the right-hand side is continuous. The unconditional solvability of this integral equation, and therefore the existence of a solution to the original problem, is deduced from the previously proven uniqueness of the solution. This is a significant point, demonstrating the close connection between the existence and uniqueness of a solution.
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