On stabilization of solutions of higher order evolution inequalities

TL;DR

This paper establishes precise conditions under which solutions to certain higher order evolution inequalities in mathematical analysis decay to zero over time, extending classical growth conditions like Keller-Osserman.

Contribution

It provides sharp, generalized criteria for the stabilization of solutions to higher order evolution inequalities, broadening the understanding beyond classical conditions.

Findings

Solutions stabilize to zero under specified growth conditions.

Conditions extend Keller-Osserman criteria to higher order inequalities.

Results apply to non-negative weak solutions in unbounded domains.

Abstract

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.