Regularity and approximation of the solution of a one-sided problemfor the Barenblatt-Zheltov-Kochina pseudoparabolic operator

We consider a one-sided problem for the Barenblatt-Zheltov-Kochina pseudoparabolic operator in the one-dimensional case, supplemented with smooth initial data and homogeneous boundary conditions. This problem is formulated in the form of a variational inequality. From the physical point of view, it models a non-stationary process of filtration of a viscous fluid in a cracky-porous gallery with a restriction on the modulus of the velocity of filtration through the cracks. The existence theorem for a weak solution of this problem is known in the literature in both one-dimensional and multidimensional cases and follows from the results obtained by M. Ptashnyk (Nonlinear Anal., 2007, vol. 66, pp. 2653-2675) using the penalty method. In M. Ptashnyk’s research, the penalty operator was chosen in a standard form, following the presentation in the monograph by J.-L. Lions “Quelques méthodes de résolution des problémes aux limites non linéaires,” Paris: Dunod Gauthier-Villars, 1969 (Theorem 5.1 in Chapter 3). In this article, we consider an approximate initial-boundary value problem for the pseudoparabolic equation incorporating Kaplan’s penalty operator and study the family of its solutions. Due to the specific structure of Kaplan’s operator, we obtain higher regularity of the weak solution of the original problem as compared to the previously known regularity properties, and also we find a strengthened property of approximating this solution by a sequence of solutions to the problem with Kaplan’s operator. In addition, we establish that the one-sided condition imposed in the original problem is satisfied by the approximate solution on a set of the spatial variable which monotone grows with decrease of the small approximation parameter.