On blow-up conditions for solutions of higher order differential inequalities

TL;DR

This paper establishes sharp conditions under which solutions to certain higher order differential inequalities must be trivial, enhancing understanding of blow-up phenomena in nonlinear differential equations.

Contribution

It provides new criteria ensuring solutions are identically zero and constructs examples demonstrating the sharpness of these conditions.

Findings

Derived conditions guaranteeing trivial solutions.

Constructed examples showing the sharpness of these conditions.

Extended understanding of blow-up behavior in higher order inequalities.

Abstract

For differential inequalities of the form $$ \sum_{|\alpha| = m} (- 1)^m \partial^\alpha a_\alpha (x, u) \ge b (x) |u|^\lambda \quad \mbox{in } {\mathbb R}^n, \: n \ge 1, $$ where $a_\alpha$ 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.