On the uniqueness of a nonlocal solution in the Barenblatt–Gilman model

This article deals with the question of uniqueness of a generalized solution to the Dirichlet–Cauchy problem for the Barenblatt–Gilman equation, which describes nonequilibrium countercurrent capillary impregnation. The unknown function corresponds to effective saturation. The main equation of this model is nonlinear and implicit with respect to the time derivative, which makes it quite hard to study. In a suitable functional space, the Dirichlet–Cauchy problem for the Barenblatt–Gilman equation reduces to the Cauchy problem for a quasilinear Sobolev-type equation. Sobolev-type equations constitute a large area of nonclassical equations of mathematical physics. The techniques used in this article originated in the theory of semilinear Sobolev-type equations. For the Cauchy problem we obtain a sufficient condition for the existence of a unique generalized solution. We establish the existence of a unique nonlocal generalized solution to the Dirichlet–Cauchy problem for the Barenblatt–Gilman equation.