arXiv subjects
Publications and source records attributed to A. E. Shishkov.
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 $ζ, ξ\in {\mathbb R}^n$.
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 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 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.
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.
For differential inequalities of the form $$ \sum_{|α| = m} (- 1)^m \partial^α a_α(x, u) \ge b (x) |u|^λ \quad \mbox{in } {\mathbb R}^n, \: n \ge 1, $$ where $a_α$ and $b$ are some functions, we obtain conditions guaranteeing that any solution is identically equal to zero. We construct examples which show that the obtained conditions are sharp.
T.I. Zelenyak's ideas and method of 1968 are shown to apply to N-dimensional second- and higher-order parabolic equtions and ensure a fast exponential convergence to degenerate equilibria. The methotology applies to autonomous and perturbed equations in Hilbert spaces.