On bow-up conditions for systems of higher order differential inequalities

TL;DR

This paper investigates conditions under which solutions to certain higher-order differential inequality systems blow up, providing precise criteria for such blow-up behavior in unbounded domains.

Contribution

It establishes exact blow-up conditions for solutions of higher-order differential inequality systems with specific growth constraints.

Findings

Derived explicit blow-up conditions for solutions

Identified critical growth thresholds for blow-up

Extended results to systems with higher-order derivatives

Abstract

We consider systems of the differential inequalities $$\left\{ \begin{aligned} & \sum_{|\alpha| = m_1} \partial^\alpha a_\alpha (x, u_1) \ge f_1 (u_2) & \mbox{in } {\mathbb R}^n, & \sum_{|\alpha| = m_2} \partial^\alpha b_\alpha (x, u_2) \ge f_2 (u_1) & \mbox{in } {\mathbb R}^n, \end{aligned} \right. $$ where $n, m_1, m_2 \ge 1$ are integers and $a_\alpha$ and $b_\alpha$ are Caratheodory functions such that $$ |a_\alpha (x, \zeta)| + |b_\alpha (x, \zeta)| \le A |\zeta| $$ with some constant $A > 0$ for almost all $x \in {\mathbb R}^n$ and for all $\zeta \in {\mathbb R}$. For solutions of these systems exact blow-up conditions are obtained.