On the stabilization of solutions of nonlinear parabolic equations with lower-order derivatives

For parabolic equations of the form
$$ \frac{\partial u}{\partial t}- \sum_{i,j=1}^n a_{ij} (x, u) \frac{\partial^2 u}{\partial x_i \partial x_j} + f (x, u, D u) = 0 in {\mathbb R}_+^{n+1}, $$
where ${\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty)$, $n \ge 1$, $D = (\partial / \partial x_1, \ldots, \partial / \partial x_n)$, and $f$ satisfies some constraints, we obtain conditions that ensure the convergence of any its solution to zero as $t \to \infty$.