DIRICHLET PROBLEM IN A HALF-SPACE FOR EQUATIONS OF THE DIFFUSION TYPE WITH INERTIA
DOI:
https://doi.org/10.31471/2304-7399-2025-21(79)-66-77Keywords:
Diffusion-type equations with inertia, potential, Poisson integral, degenerate parabolic equations, Dirichlet problem, degenerate parabolic equations, diffusion processes.Abstract
The article investigates the first boundary value problem in a half-space for an equation of the diffusion type with inertia, where inertia depends on four groups of variables with degeneracy of parabolicity, each group is subordinated to a certain number of variables. To establish the existence of solutions of boundary value problems, the limit transition and asymptotic expansion of the equation are used. We investigate the first boundary value problem in a half-space for a degenerate parabolic equation of the diffusion type with inertia. We use the explicit form of the fundamental solution of the Cauchy problem and the properties of its derivatives for a degenerate parabolic equation, and by the method of potentials we reduce the first boundary value problem in a half-space to the solution of the corresponding singular integral equation in classes of exponentially decreasing differential functions, such that the integral operator is a compression operator for small t. The uniqueness of the solution of the first boundary value problem in a half-space follows from the maximum principle in classes of bounded functions. In this article we have established the existence and uniqueness of the solution of the Dirichlet problem for one class of degenerate equations of the diffusion type with inertia in the case of three groups of degenerate variables, by the method of potentials. The uniqueness of the solution of the problem follows from the maximum principle in classes of bounded functions.
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