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

Abstract

In this paper we study the well-posedness of the evolution equation of the form $u^{\prime }(t)=Au(t)+Cu(t)$, $t\ge 0$, 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 ${\left(T_A(t)\right)}_{t\ge 0}$ with $\Vert T(t)\Vert \le M{e}^{\omega t}$ in a Banach space $\mathbb{X}$ and $C$ is a linear operator in $\mathbb{X}$ such that $D(A)\subset D(C)$ and $\Vert CR(\mu ,A)\Vert \le K/(\mu -\omega )$ for each $\mu >\omega$, then, the above-mentioned evolution equation is well-posed, that is, $A+C$ generates a $C_0$-semigroup ${\left(T_{A+C}(t)\right)}_{t\ge 0}$ satisfying $\Vert T_{A+C}(t)\Vert \le M{e}^{(\omega +MK)t}$. 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.

12 pages