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On the uniqueness of a nonlocal solution in the Barenblatt-Gilman model. (Russian. English summary) Zbl 1329.47085

Summary: 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 eective 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 sucient 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.

MSC:

47N50 Applications of operator theory in the physical sciences
76D45 Capillarity (surface tension) for incompressible viscous fluids
47J05 Equations involving nonlinear operators (general)
76S05 Flows in porous media; filtration; seepage
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