arXiv subjects
Publications and source records attributed to A. A. Kon'kov.
We study solutions of the differential inequality $$ Δ^{m / 2} u \ge f (x) g (u) \quad \mbox{in } B_1 \setminus \{ 0 \}, $$ where $m \ge 2$ is an even integer, $f$ and $g$ are some functions, and $B_1$ is an open unit ball in $R^n$, $n \ge 2$, centered at zero. Our aim is to obtain a necessary condition for a singularity at zero to be removable for any solution of this inequality.
We study systems of the differential inequalities $$ \left\{ \begin{aligned} & - \operatorname{div} A_1 (x, \nabla u_1) \ge F_1 (x, u_2) & \mbox{in } {\mathbb R}^n, & - \operatorname{div} A_2 (x, \nabla u_2) \ge F_2 (x, u_1) & \mbox{in } {\mathbb R}^n, \end{aligned} \right. $$ where $n \ge 2$ and $A_i$ are Caratheodory functions such that $$ C_1 |ξ|^{p_i} \le ξ A_i (x, ξ), \quad |A_i (x, ξ)| \le C_2 |ξ|^{p_i - 1}, \quad i = 1,2, $$ with some constants $C_1, C_2 > 0$ and $p_1, p_2 > 1$ for almost all $x \in {\mathbb R}^n$ and for all $ξ\in {\mathbb R}^n$, $n \ge 2$. For non-negative solutions of these systems we obtain exact blow-up conditions.
We consider systems of the differential inequalities $$\left\{ \begin{aligned} & \sum_{|α| = m_1} \partial^αa_α(x, u_1) \ge f_1 (u_2) & \mbox{in } {\mathbb R}^n, & \sum_{|α| = m_2} \partial^αb_α(x, u_2) \ge f_2 (u_1) & \mbox{in } {\mathbb R}^n, \end{aligned} \right. $$ where $n, m_1, m_2 \ge 1$ are integers and $a_α$ and $b_α$ are Caratheodory functions such that $$ |a_α(x, ζ)| + |b_α(x, ζ)| \le A |ζ| $$ with some constant $A > 0$ for almost all $x \in {\mathbb R}^n$ and for all $ζ\in {\mathbb R}$. For solutions of these systems exact blow-up conditions are obtained.
We study the existence of global positive solutions of the differential inequalities $$ - \operatorname{div} A (x, u, \nabla u) \ge f (u) \quad \mbox{in } {\mathbb R}^n, $$ where $n \ge 2$ and $A$ is a Carathéodory function such that $$ (A (x, s, ζ) - A (x, s, ξ))(ζ- ξ) \ge 0, $$ $$ C_1 |ξ|^p \le ξ A (x, s, ξ), \quad |A (x, s, ξ)| \le C_2 |ξ|^{p-1}, \quad C_1, C_2 > 0, \; p > 1, $$ for almost all $x \in {\mathbb R}^n$ and for all $s \in {\mathbb R}$ and $ζ, ξ\in {\mathbb R}^n$.
We show the exactness of the Kiguradze--Kvinikadze blow-up condition for solutions of the Cauchy problem $$ w^{(m)} = f (t, w, \ldots, w^{(m - 1)}), \qquad w^{(i)} (0) = a_i \ge 0, \quad i = 0, \ldots, m - 1. $$
For the problem $$ \left\{ \begin{aligned} & \partial_t^k u - \sum_{|α| = m} \partial^α a_α(x, t, u) \ge f (|u|) \quad \mbox{in } {\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty), & u (x, 0) = u_0 (x), \: \partial_t u (x, 0) = u_1 (x), \ldots, \partial_t^{k-1} u (x, 0) = u_{k-1} (x) \ge 0, \end{aligned} \right. $$ we obtain exact conditions on the function $f$ guaranteeing that any global weak solution is identically zero.
We consider the inequality $$ - \operatorname{div} A (x, \nabla u) \ge f (u) \quad \mbox{in } {\mathbb R}^n, $$ where $n \ge 2$ and $A$ is a Caratheodory function such that $$ C_1 |ξ|^p \le ξ A (x, ξ) \quad \mbox{and} \quad |A (x, ξ)| \le C_2 |ξ|^{p-1} $$ with some constants $C_1 > 0$, $C_2 > 0$, and $p > 1$ for almost all $x \in {\mathbb R}^n$ and for all $ξ\in {\mathbb R}^n$. Our aim is to find exact conditions on the function $f$ guaranteeing that any non-negative solution of this inequality is identically zero.
We obtain a Dini type blow-up condition for global weak solutions of the differential inequality $$ \sum_{|α| = m} \partial^α a_α(x, u) \ge g (|u|) \quad \mbox{in } {\mathbb R}^n, $$ where $m, n \ge 1$ are integers and $a_α$ and $g$ are some functions.
We obtain a criterion for the existence of solutions of the problem $$ Δ_p u = 0 \quad \mbox{in } M \setminus \partial M, \quad \left. u \right|_{ \partial M } = h, $$ with the bounded Dirichlet integral, where $M$ is an oriented complete Riemannian manifold with boundary and $h \in W_{p, loc}^1 (M)$, $p > 1$.
For differential inequalities with the $\infty$-Laplacian in the principal part, we obtain conditions for the absence of solutions in unbounded domains. Examples are given to demonstrate the accuracy of these conditions.
Let $Ω\ne \emptyset$ be an unbounded open subset of ${\mathbb R}^n$, $n \ge 2$. We obtain blow-up conditions for non-negative solutions of the problem $$ {\mathcal L}_φu \ge F (x, u) \quad \mbox{in } Ω, \quad \left. u \right|_{ \partial Ω } = 0, $$ where $φ$ and $F$ are some function and $$ {\mathcal L}_φu = \operatorname{div} \left( \frac{ φ(|\nabla u|) }{ |\nabla u| } \nabla u \right) $$ is the $φ$-Laplace operator.
We obtain conditions guaranteeing that weak solutions of the differential inequality $$ \sum_{|α| = m} \partial^α a_α(x, u) \ge f (x) g (|u|) \quad \mbox{in } Ω\setminus S, $$ has a removable singular set $S \subset Ω$, where $Ω$ is a bounded domain and $a_α$, $f$, and $g$ are some functions.
We obtain necessary and sufficient existence conditions for solutions of the boundary value problem $$ Δ_p u = f \quad \mbox{on } M, \quad \left. \left| \nabla u \right|^{p - 2} \frac{\partial u}{\partial ν} \right|_{ \partial M } = h, $$ where $p > 1$ is a real number, $M$ is a connected oriented complete Riemannian manifold with boundary, and $ν$ is the external normal vector to $\partial M$.
We obtain stabilization conditions and large time estimates for weak solutions of the inequality $$ \sum_{|α| = m} \partial^α a_α(x, t, u) - u_t \ge f (x, t) g (u) \quad \mbox{in } Ω\times (0, \infty), $$ where $Ω$ is a non-empty open subset of ${\mathbb R}^n$, $m, n \ge 1$, and $a_α$ are Caratheodory functions such that $$ |a_α(x, t, ζ)| \le A ζ^p, \quad |α| = m, $$ with some constants $A > 0$ and $0 < p < 1$ for almost all $(x, t) \in Ω\times (0, \infty)$ and for all $ζ\in [0, \infty)$. For solutions of homogeneous differential inequalities, we give an exact universal upper bound.
We obtain sufficient conditions for solutions of the $m$th-order differential inequality $$ \sum_{|α| = m} \partial^αa_α(x, u) \ge f (x) g (|u|) \quad \mbox{in } B_1 \setminus \{ 0 \} $$ to have a removable singularity at zero, where $a_α$, $f$, and $g$ are some functions, and $B_1 = \{ x : |x| < 1 \}$ is a unit ball in ${\mathbb R}^n$. Constructed examples demonstrate the exactness of these conditions.
Assume that $p > 1$ and $p - 1 \le α\le p$ are real numbers and $Ω$ is a non-empty open subset of ${\mathbb R}^n$, $n \ge 2$. We consider the inequality $$ {\rm div} \, A (x, D u) + b (x) |D u|^α \ge 0, $$ where $D = (\partial / \partial x_1, \ldots, \partial / \partial x_n)$ is the gradient operator and $A : Ω\times {\mathbb R}^n \to {\mathbb R}^n$ and $b : Ω\to [0, \infty)$ are some functions with $$ C_1 |ξ|^p \le ξ A (x, ξ), \quad |A (x, ξ)| \le C_2 |ξ|^{p-1}, \quad C_1, C_2 = const > 0, $$ for almost all $x \in Ω$ and for all $ξ\in {\mathbb R}^n$. For solutions of this inequality we obtain estimates depending on the geometry of $Ω$. In particular, these estimates imply regularity conditions of a boundary point.
We obtain exact conditions guaranteeing that any global weak solution of the differential inequality $$ \sum_{|α| = m} \partial^α a_α(x, u) \ge g (|u|) \quad \mbox{in } {\mathbb R}^n $$ is trivial, where $m, n \ge 1$ are integers and $a_α$ and $g$ are some functions. These conditions generalize the well-know Keller-Osserman condition.
We obtain sharp conditions guaranteeing that every non-negative weak solution of the inequality $$ \sum_{|α| = m} \partial^α a_α(x, t, u) - u_t \ge f (x, t) g (u) \quad \mbox{in} {\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty), \quad m,n \ge 1, $$ stabilizes to zero as $t \to \infty$. These conditions generalize the well-known Keller-Osserman condition on the grows of the function $g$ at infinity.