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

Abstract

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 $a_{\alpha }$ 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.