Estimates of the stabilization rate as $t\to\infty$ of solutions of the first mixed problem for a quasilinear system of second-order parabolic equations

A quasilinear system of parabolic equations with energy inequality is considered in a cylindrical domain $\{t>0\}\times\Omega$. In a broad class of unbounded domains $\Omega$ two geometric characteristics of a domain are identified which determine the rate of convergence to zero as $t\to\infty$ of the $L_2$-norm of a solution. Under additional assumptions on the coefficients of the quasilinear system estimates of the derivatives and uniform estimates of the solution are obtained; they are proved to be best possible in the order of convergence to zero in the case of one semilinear equation.