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

TL;DR

This paper investigates the long-term behavior of solutions to certain higher-order evolution inequalities with fast diffusion, establishing stabilization conditions, large-time estimates, and an exact universal upper bound for homogeneous cases.

Contribution

It provides new stabilization criteria and explicit large-time bounds for weak solutions of higher-order inequalities with fast diffusion, extending existing theory.

Findings

Established stabilization conditions for solutions.

Derived large-time estimates for weak solutions.

Provided an exact universal upper bound for homogeneous inequalities.

Abstract

We obtain stabilization conditions and large time estimates for weak solutions 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 open subset of ${\mathbb R}^n$, $m, n \ge 1$, and $a_\alpha$ are Caratheodory functions 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 $\zeta \in [0, \infty)$. For solutions of homogeneous differential inequalities, we give an exact universal upper bound.