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The phase space of the Cauchy-Dirichlet problem for the Oskolkov equation of nonlinear filtration. (English. Russian original) Zbl 1076.35064
Russ. Math. 47, No. 9, 33-38 (2003); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2003, No. 9, 36-41 (2003).
From the text: Let \(\Omega\subset\mathbb{R}^n\) be a bounded domain with a boundary \(\partial\Omega\) of the class \(C^\infty\). In the domain \(\Omega\times\mathbb{R}\), the following Cauchy-Dirichlet problem is considered: \[ u(x,0)= u_0(x),\quad x\in\Omega;\qquad u(x,t)= 0,\quad (x,t)\in\partial\Omega\times \mathbb{R}\tag{1} \] for the nonclassical partial differential equation \[ (1- \kappa\Delta)u_t= \nu\Delta u- |u|^{q-2} u+ f,\quad q\geq 2.\tag{2} \] Equation (2) describes the dynamics of pressure of a non-Newtonian fluid, which is filtered in a porous medium. The goal of this article is the investigation of the solvability of the problem (1)–(2) under the conditions \(\kappa\in\mathbb{R}\setminus\{0\}\) and \(\nu\in\mathbb{R}_+\).

MSC:
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35K55 Nonlinear parabolic equations
76S05 Flows in porous media; filtration; seepage
76A05 Non-Newtonian fluids
35Q35 PDEs in connection with fluid mechanics
35K90 Abstract parabolic equations
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