Solutions of anisotropic parabolic equations with double non-linearity in unbounded domains

This work is devoted to some class of parabolic equations of high order with double nonlinearity which can be represented by a model equation
$$\begin{gather*} \frac{\partial}{\partial t}(|u|^{k-2}u)= \sum_{\alpha=1}^n(-1)^{m_\alpha-1}\frac{\partial^{m_\alpha}}{\partial x_\alpha^{m_\alpha}} \left[\left|\frac{\partial^{m_\alpha} u}{\partial x_\alpha^{m_\alpha}}\right|^{p_\alpha-2} \frac{\partial^{m_\alpha} u}{\partial x_\alpha^{m_\alpha}}\right],\\ m_1,\ldots, m_n\in \mathbb{N},\quad p_n\geq \ldots \geq p_1>k,\quad k>1. \end{gather*}$$
For the solution of the first mixed problem in a cylindrical domain $D=(0,\infty)$ $\times\Omega, \;\Omega\subset \mathbb{R}_n,$ $n\geq 2,$ with homogeneous Dirichlet boundary condition and finite initial function the highest rate of decay established as $t \to \infty$. Earlier upper estimates were obtained by the authors for anisotropic equation of the second order and prove their accuracy.