A generation theorem for the perturbation of strongly continuous semigroups by unbounded operators

In this paper we study the well-posedness of the evolution equation of the form $u'(t)=Au(t)+Cu(t)$, $t\ge 0$, where $A$ is the generator of a $C_0$- semigroup and $C$ is a (possibly unbounded) linear operator in a Banach space $\mathbb{X}$. We prove that if $A$ generates a $C_0$-semigroup $\left (T_A(t)\right )_{t \geq 0}$ with $\|T(t)\| \le Me^{\omega t}$ in a Banach space $\mathbb{X}$ and $C$ is a linear operator in $\mathbb{X}$ such that $D(A)\subset D(C)$ and $\| CR(\mu ,A)\| \le K/(\mu -\omega)$ for each $\mu>\omega$, then, the above-mentioned evolution equation is well-posed, that is, $A+C$ generates a $C_0$-semigroup $\left (T_{A+C}(t)\right )_{t \geq 0}$ satisfying $\| T_{A+C}(t)\| \le Me^{(\omega +MK)t}$. Our approach is to use the Hille-Yosida Theorem. Discussions on the persistence of asymptotic behavior of the perturbed equations such as the roughness of exponential dichotomy are also given. The obtained results seem to be new.