Geodesic
Stabilization of the solution of a~doubly nonlinear parabolic equation
Sbornik. Mathematics, Tome 204 (2013) no. 9, pp. 1239-1263

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The method of Galerkin approximations is employed to prove the existence of a strong global (in time) solution of a doubly nonlinear parabolic equation in an unbounded domain. The second integral identity is established for Galerkin approximations, and passing to the limit in it an estimate for the decay rate of the norm of the solution from below is obtained. The estimates characterizing the decay rate of the solution as $x\to\infty$ obtained here are used to derive an upper bound for the decay rate of the solution with respect to time; the resulting estimate is pretty close to the lower one. Bibliography: 17 titles.
Keywords: doubly nonlinear parabolic equation, rate of decay of the solution, lower estimate, existence of a strong global (in time) solution. 
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È. R. Andriyanova; F. Kh. Mukminov. Stabilization of the solution of a~doubly nonlinear parabolic equation. Sbornik. Mathematics, Tome 204 (2013) no. 9, pp. 1239-1263. http://geodesic.mathdoc.fr/item/SM_2013_204_9_a0/