On stabilization of solutions of higher order evolution inequalities

Abstract

We obtain sharp conditions guaranteeing that every non-negative weak solution of the inequality $$\begin{array}{c}{\displaystyle \sum }_{|\mathit{\alpha }|=\mathit{m}}{\partial }^{\mathit{\alpha }}{\mathit{a}}_{\mathit{\alpha }}\mathrm{(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{,}\mathit{u}\mathrm{)}-{\mathit{u}}_{\mathit{t}}⩾\mathit{f}\mathrm{(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{)}\mathit{g}\mathrm{(}\mathit{u}\mathrm{)}\text{in }{\mathbb{R}}_{+}^{\mathit{n}+1}={\mathbb{R}}^{\mathit{n}}\times \mathrm{(}0\mathrm{,}\infty \mathrm{)}\mathrm{,}\mathit{m}\mathrm{,}\mathit{n}⩾1\mathrm{,}\end{array}$$ stabilizes to zero as $\mathit{t}\to \infty$ . These conditions generalize the well-known Keller–Osserman condition on the growth of the function g at infinity.