Geodesic
Solutions to higher-order anisotropic parabolic equations in unbounded domains
Sbornik. Mathematics, Tome 205 (2014) no. 1, pp. 7-44

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The paper is devoted to a certain class of doubly nonlinear higher-order anisotropic parabolic equations. Using Galerkin approximations it is proved that the first mixed problem with homogeneous Dirichlet boundary condition has a strong solution in the cylinder $D=(0,\infty)\times\Omega$, where $\Omega\subset\mathbb R^n$, $n\geqslant 3$, is an unbounded domain. When the initial function has compact support the highest possible rate of decay of this solution as $t\to \infty$ is found. An upper estimate characterizing the decay of the solution is established, which is close to the lower estimate if the domain is sufficiently ‘narrow’. The same authors have previously obtained results of this type for second order anisotropic parabolic equations. Bibliography: 29 titles.
Keywords: higher-order anisotropic equation, parabolic equation with double nonlinearity, rate of decay of a solution.
Mots-clés : existence of a solution 
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L. M. Kozhevnikova; A. A. Leont'ev. Solutions to higher-order anisotropic parabolic equations in unbounded domains. Sbornik. Mathematics, Tome 205 (2014) no. 1, pp. 7-44. http://geodesic.mathdoc.fr/item/SM_2014_205_1_a1/