On stabilization of solutions of nonlinear parabolic equations with a gradient term

Abstract

For parabolic equations of the form $$ \frac{\partial u}{\partial t} - \sum_{i,j=1}^n a_{ij} (x, u) \frac{\partial^2 u}{\partial x_i \partial x_j} + f (x, u, D u) = 0 \quad \mbox{in } {\mathbb R}_+^{n+1}, $$ where ${\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty)$, $n \ge 1$, $D = (\partial / \partial x_1, \ldots, \partial / \partial x_n)$ is the gradient operator, and $f$ is some function, we obtain conditions guaranteeing that every solution tends to zero as $t \to \infty$.