On the well-posedness of linear evolution equations under unbounded nonautonomous perturbations

TL;DR

This paper establishes conditions for the well-posedness of nonautonomous linear evolution equations with unbounded perturbations, focusing on the properties of the perturbation operators and their impact on the existence and uniqueness of solutions.

Contribution

It introduces a framework for analyzing well-posedness of evolution equations with unbounded, time-dependent perturbations using a specific operator norm and differentiability conditions.

Findings

Existence of an evolution family under continuity of (t) in the operator norm.

Uniqueness of the evolution family when (t) is continuously differentiable.

Examples demonstrating the applicability of the theoretical results.

Abstract

We study conditions for the well-posedness of nonautonomous perturbation of evolution equations of the form \[ u'(t)=(A+B(t))u(t), \quad t \in [a,b], \] where $A$ generates a $\mathrm{C}_0$-semigroup $\left (T(t)\right )_{t\ge 0}$ with $\| T(t)\| \le Me^{\omega_0 t}$, $t\ge 0$, in a Banach space $\mathbb{X}$ and $B(t)$ are $t$-dependent (unbounded) linear operators in $\mathbb{X}$. The unbounded perturbation operators $B(t)$ are assumed to belong to a normed space (denoted by $\mathcal{GL}_A (\mathbb{X})$) of unbounded linear operators $C$ in $\mathbb{X}$ such that $D(A) \subset D(C)$ with norm \[ \| C\|_A:= (1/M) \sup_{\mu >\omega_0 } \| (\mu-\omega_0) CR(\mu,A)\| <\infty. \] We prove that the above-mentioned evolution equation admits an evolution family if $\| B(\cdot)\|_A$ is continuous in $[a,b]$. The evolution family is unique if $B(\cdot)R(\mu, A)$ as a function $[a,b]\to \mathcal{L}(\mathbb{X})$ is continuously differentiable, and \[ \limsup_{\mu \to\infty} \sup_{t\in [a,b]} \left \| \frac{d}{dt}[B(t)R(\mu,A)]\right \| <\infty. \] Examples are given to illustrate the obtained results.