Stabilization of solutions of an anisotropic quasilinear parabolic equation in unbounded domains

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.