Estimates of solutions of an anisotropic doubly nonlinear parabolic equation

The first mixed problem with the Dirihlet homogeneous boundary-value condition and a finite initial function is considered for a certain class of second-order anisotropic doubly nonlinear parabolic equations in a cylindrical domain $D=(0,\infty)\times\Omega$. Upper estimates characterizing the dependence of the decay rate of the solution to the problem on geometry of an unbounded domain $\Omega\subset\mathbb R_n$, $n\geq3$, are established when $t\to\infty$. Existence of strong solutions is proved by the method of Galerkin's approximations. The method of their construction for the modelling isotropic equation has been earlier offered by F. Kh. Mukminov, E. R. Andriyanova. The estimate of the admissible decay rate of the solution on an unbounded domain has been obtained on the basis of Galerkin's approximations. It proves the accuracy of the upper estimate.