Eventual Domination for Linear Evolution Equations

TL;DR

This paper establishes criteria for when the orbits of one semigroup eventually dominate those of another on Banach lattices, with applications to differential operators, providing new insights into their long-term behavior.

Contribution

It offers necessary and sufficient conditions for orbit domination between semigroups, including criteria for self-adjoint operators, advancing understanding of their asymptotic dominance.

Findings

Criteria for orbit domination in Banach lattices

Conditions for eventual domination of semigroup orbits

Insights into differential operators' long-term behavior

Abstract

We consider two $C_0$-semigroups on function spaces or, more generally, Banach lattices and give necessary and sufficient conditions for the orbits of the first semigroup to dominate the orbits of the second semigroup for large times. As an important special case we consider an $L^2$-space and self-adjoint operators $A$ and $B$ which generate $C_0$-semigroups; in this situation we give criteria for the existence of a time $t_1 \ge 0$ such that $e^{tB} \ge e^{tA}$ for all subsequent times $t\ge t_1$. As a consequence of our abstract theory, we obtain many surprising insights into the behaviour of various second and fourth order differential operators.