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We obtain stabilization conditions and large time estimates for weaksolutions of the inequality $$  \sum_{|\alpha| = m}  \partial^\alpha  a_\alpha (x, t, u)  -  u_t  \ge  f (x, t) g (u)  \quad  \mbox{in } \Omega \times (0, \infty), $$ where $\Omega$ is a non-empty opensubset of ${\mathbb R}^n$, $m, n \ge 1$, and $a_\alpha$ are Caratheodoryfunctions such that $$  |a_\alpha (x, t, \zeta)|  \le  A \zeta^p,  \quad  |\alpha| = m, $$ with some constants $A > 0$ and $0 < p < 1$ for almost all$(x, t) \in \Omega \times (0, \infty)$ and for all $\zeta \in [0, \infty)$. Forsolutions of homogeneous differential inequalities, we give an exact universalupper bound.

On large time behavior of solutions of higher order evolution inequalities with fast diffusion