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,
aΞ±β and f are some functions, and u0ββL1,locβ(Rn)