Stabilization of solutions of quasilinear second order parabolic equations in domains with non-compact boundaries

The first mixed problem with homogeneous Dirichlet boundary condition and initial function with compact support is considered for quasilinear second order parabolic equations in a cylindrical domain $D=(0,\infty)\times\Omega$. Upper bounds are obtained, which give the rate of decay of the solutions as $t\to\infty$ as a function of the geometry of the unbounded domain $\Omega\subset \mathbb R_n$, $n\geqslant 2$.
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