The third boundary value problem for the system of equations of non-equilibrium sorption

In this paper, we investigate the system of equations modeling the process of non-equilibrium sorption. In particular, such systems are used in mathematical modeling of the production process of the useful component by the method of borehole underground leaching. The theorem of existence and uniqueness of the solution of the third boundary value problem in the multidimensional case in Hölder classes of functions is proved. The obtained maximum principle plays an important role in the proof of the theorem. The uniqueness of the solution is a consequence of this principle. The existence of a solution to the problem is shown by Schauder's fixed point theorem of a completely continuous operator. The description of the corresponding operator is given. Estimates are obtained to ensure the continuity of the constructed operator, and it is shown that the operator maps the original set of functions into itself at a small period of time. Then the estimates are given, allowing to continue the solution to any finite value of time.