On a Solution of the Dirichlet–Cauchy Problem for the Barenblatt–Gilman Equation

We investigate the solvability of the Dirichlet–Cauchy problem for the Barenblatt–Gilman equation modeling the nonequilibrium countercurrent capillary impregnation. The feature of this model is the consideration of non-equilibrium effect — this becomes especially important when the process of impregnation takes a long time. Irregular and complex structure of the pore space does not allow to study the movement of liquids and gases therein by conventional methods of hydrodynamics. Hence the design and analysis of specific models describing these processes are required. The main equation of the model is nonlinear and not solvable for the derivative. This creates a significant difficulty in its consideration. The authors attribute the Barenblatt – Gilman equation to the wide class of Sobolev type equations. Sobolev type equations constitute an extensive area of nonclassical equations of mathematical physics. Research methods that are used in the work are initially emerged in the theory of semilinear Sobolev type equations. The equation is first considered in this context. The original problem is solved by the reduction in suitable functional spaces to the Cauchy problem for an abstract quasilinear Sobolev type equation with $s$-monotone and $p$-coercive operator. Existence theorems have been proven for generalized solutions of the abstract and the original problem.