Manakova, Natal’ya Aleksandrovna; Bogatyreva, Ekaterina Aleksandrovna On a solution of the Dirichlet-Cauchy problem for the Barenblatt-Gilman equation. (Russian. English summary) Zbl 1330.47096 Izv. Irkutsk. Gos. Univ., Ser. Mat. 7, 52-60 (2014). Summary: 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. Cited in 2 Documents MSC: 47N20 Applications of operator theory to differential and integral equations 76S05 Flows in porous media; filtration; seepage 76D45 Capillarity (surface tension) for incompressible viscous fluids 35Q35 PDEs in connection with fluid mechanics Keywords:Barenblatt-Gilman equation; countercurrent capillary impregnation; quaslinear Sobolev type equation PDFBibTeX XMLCite \textit{N. A. Manakova} and \textit{E. A. Bogatyreva}, Izv. Irkutsk. Gos. Univ., Ser. Mat. 7, 52--60 (2014; Zbl 1330.47096) Full Text: MNR