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

We study conditions for the well-posedness of nonautonomous perturbation of evolution equations of the form $$u^{\mathrm{\prime }}(t)=(A+B(t))u(t),t\in [a,b],$$ where $A$ generates a ${\mathrm{C}}_0$-semigroup ${(T(t))}_{t\ge 0}$ with $\mathrm{\parallel }T(t)\mathrm{\parallel }\le M{e}^{{\omega }_{0}t}$, $t\ge 0$, 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 ${\mathcal{G}\mathcal{L}}_A(\mathbb{X})$) of unbounded linear operators $C$ in $\mathbb{X}$ such that $D(A)\subset D(C)$ with norm $$\mathrm{\parallel }C{\mathrm{\parallel }}_A:=(1\mathrm{/}M)\underset{\mu >{\omega }_0}{\sup }\mathrm{\parallel }(\mu -{\omega }_0)CR(\mu ,A)\mathrm{\parallel }<\mathrm{\infty }\mathrm{.}$$ We prove that the above-mentioned evolution equation admits an evolution family if $\mathrm{\parallel }B(\cdot ){\mathrm{\parallel }}_A$ 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 $$\underset{\mu \to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\underset{t\in [a,b]}{\sup }\parallel \frac{d}{dt}[B(t)R(\mu ,A)]\parallel <\mathrm{\infty }\mathrm{.}$$ Examples are given to illustrate the obtained results.