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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