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Andrej A. Kon'kov

Publications and source records attributed to Andrej A. Kon'kov.

5 recordsLinked to original sources
arXiv2017

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

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 $t \to \infty$.

math.AP↗
arXiv2014

On properties of solutions of quasilinear second-order elliptic inequalities

Let $Ω$ be an unbounded open subset of ${\mathbb R}^n$, $n \ge 2$, and $A : Ω\times {\mathbb R}^n \to {\mathbb R}^n$ be a function such that $$ C_1 |ζ|^p \le ζ A (x, ζ), \quad |A (x, ζ)| \le C_2 |ζ|^{p-1} $$ with some constants $C_1 > 0$, $C_2 > 0$, and $p > 1$ for almost all $x \in Ω$ and for all $ζ\in {\mathbb R}^n$. We obtain blow-up conditions and priori estimates for inequalities of the form $$ {\rm div} \, A (x, D u) + b (x) |D u|^α \ge q (x) g (u) \quad \mbox{in } Ω, $$ where $p - 1 \le α\le p$ is a real number and, moreover, $b \in L_{\infty, loc} (Ω)$, $q \in L_{\infty, loc} (Ω)$, and $g \in C ([0, \infty))$ are non-negative functions.

math.AP↗