On blow-up conditions for nonlinear higher order evolution inequalities

Abstract

We obtain exact conditions for global weak solutions of the problem \left\{ \begin{aligned} & u_t - \sum_{|\alpha| = m} \partial^\alpha a_\alpha (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) \ge 0, \end{aligned} \right. to be identically zero, where $m$ and $n$ are positive integers, $a_\alpha$ and $f$ are some functions, and $u_0 \in L_{1, loc} ({\mathbb R}^n)$