On The Stabilization of Solutions of Nonlinear Parabolic Equations with Lower-Order Derivatives

Abstract

For parabolic equations of the form

$$ \frac{\partial u}{\partial t}-\sum \limits_{i,j=1}^n{a}_{ij}\left(x,u\right)\frac{\partial^2u}{\partial {x}_i\partial {x}_j}+f\left(x,u, Du\right)=0\kern0.5em \mathrm{in}\kern0.5em {\mathbb{R}}_{+}^{n+1}, $$

where ${\displaystyle \begin{array}{cc}{\mathbb{R}}_{+}^{n+1}={\mathbb{R}}^n\times \left(0,\infty \right),& n\ge 1,D=\Big(\partial \end{array}}/\partial {x}_1,...,\partial /\partial {x}_n\Big),$ and f satisfies some constraints, we obtain conditions that ensure the convergence of any its solution to zero as t→∞.