For the problem \(\begin{aligned} \left\{ \begin{aligned}&\partial _t^k u - \sum _{|\alpha | = m} \partial ^\alpha a_\alpha (x, t, u) \ge f (|u|) \quad \text{ in } {\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty ), \\&u (x, 0) = u_0 (x), \, \partial _t u (x, 0) = u_1 (x), \ldots , \partial _t^{k-1} u (x, 0) = u_{k-1} (x) \ge 0, \end{aligned} \right. \end{aligned}\) where \(u_i \in L_{1, loc} ({\mathbb R}^n)\) and \(a_\alpha \) are Caratheodory functions such that \(\begin{aligned} |a_\alpha (x, t, \zeta )| \le A |\zeta |^p, \quad A, p = const > 0, \end{aligned}\) for almost all \((x, t) \in {\mathbb R}_+^{n+1}\) and for all \(\zeta \in {\mathbb R}\) , we obtain exact conditions on the function f guaranteeing that any global weak solution is identically zero.
