Geodesic
Stabilization of solutions of an anisotropic quasilinear parabolic equation in unbounded domains
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 278 (2012), pp. 114-128

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The first initial-boundary value problem with the homogeneous Dirichlet boundary condition and a compactly supported initial function is considered for a model second-order anisotropic parabolic equation in a cylindrical domain $D=(0,\infty)\times\Omega$. We find an upper bound that characterizes the dependence of the decay rate of solutions as $t\to\infty$ on the geometry of the unbounded domain $\Omega\subset\mathbb R_n$, $n\geq3$, and on nonlinearity exponents. We also obtain an estimate for the admissible decay rate of nonnegative solutions in unbounded domains; this estimate shows that the upper bound is sharp. 
@article{TM_2012_278_a10,
     author = {L. M. Kozhevnikova and F. Kh. Mukminov},
     title = {Stabilization of solutions of an anisotropic quasilinear parabolic equation in unbounded domains},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {114--128},
     publisher = {mathdoc},
     volume = {278},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2012_278_a10/}
}
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L. M. Kozhevnikova; F. Kh. Mukminov. Stabilization of solutions of an anisotropic quasilinear parabolic equation in unbounded domains. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 278 (2012), pp. 114-128. http://geodesic.mathdoc.fr/item/TM_2012_278_a10/