Estimate of the rate of stabilization of the solution of the first initial-boundary problem for the equation of a porous medium in an unbounded region (Q1907437)

The stabilization of the solution of the Cauchy-Dirichlet problem for the porous medium equation \[ u_t= \Delta u^m,\quad (x, t)\in D= \mathbb{R}^n_k\times \{t> 0\},\tag{1} \] with initial boundary conditions of the form \[ (2)\quad u(x, t)= 0,\;(x, t)\in \partial\mathbb{R}^n_k\times \{t> 0\},\qquad (3)\quad u(x, 0)= u_0(x),\;x\in \mathbb{R}^n_k, \] where \(m> 1\), \(n\geq 1\), \(\mathbb{R}^n_k= \mathbb{R}^n\cap \{x_1,\dots, x_k> 0\}\) is studied. The following assertion is a principal result of the paper: Let \(u(x, t)\) be a solution of (1)--(3) in \(D\) and \(\mu_k(t)\equiv \int_{\mathbb{R}^n_k} \overline x_k u(x, t) dx< \infty\), then there exist positive constants \(C\), \(c\) and \(t_0\) such that the following estimates are valid: \[ M(t)\equiv \text{ess}\sup_{\mathbb{R}^n_k} u(x, t)\leq C(\mu_k(0))^{2\sigma/(n+ k)} t^{-\sigma},\;t> 0,\;M(t)\geq c\mu_k(0) t^{- \sigma},\;t> t_0. \]