Uniform convergence of operator semigroups without time regularity

TL;DR

This paper establishes new criteria for the operator norm convergence of general semigroups without assuming time regularity, extending analysis tools to non-strongly continuous semigroups like those arising in certain parabolic equations.

Contribution

It introduces a unified approach to characterize operator semigroup convergence without time regularity, applicable to both continuous and discrete cases, using the concept of the 'semigroup at infinity.'

Findings

Provides new convergence criteria for non-strongly continuous semigroups.

Extends analysis techniques to discrete-time and abstract semigroup representations.

Proves a convergence theorem for solutions to certain parabolic systems.

Abstract

When we are interested in the long-term behaviour of solutions to linear evolution equations, a large variety of techniques from the theory of $C_0$-semigroups is at our disposal. However, if we consider for instance parabolic equations with unbounded coefficients on $\mathbb{R}^d$, the solution semigroup will not be strongly continuous, in general. For such semigroups many tools that can be used to investigate the asymptotic behaviour of $C_0$-semigroups are not available anymore and, hence, much less is known about their long-time behaviour. Motivated by this observation, we prove new characterisations of the operator norm convergence of general semigroup representations - without any time regularity assumptions - by adapting the concept of the "semigroup at infinity", recently introduced by M.~Haase and the second named author. Besides its independence of time regularity, our approach also allows us to treat the discrete-time case (i.e., powers of a single operator) and even more abstract semigroup representations within the same unified setting. As an application of our results, we prove a convergence theorem for solutions to systems of parabolic equations with the aforementioned properties.