  We obtain sharp conditions guaranteeing that every non-negative weak solution
of the inequality $$ \sum_{|\alpha| = m}
  \partial^\alpha
  a_\alpha (x, t, u)
  -
  u_t
  \ge
  f (x, t) g (u)
  \quad
  \mbox{in} {\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty),
  \quad
  m,n \ge 1, $$ stabilizes to zero as $t \to \infty$. These conditions
generalize the well-known Keller-Osserman condition on the grows of the
function $g$ at infinity.
