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≥ 0, where A is the generator of a C₀- semigroup and C is a (possibly unbounded) linear operator in a Banach space X. We prove that if A generates a C₀-semigroup (TA(t) )t ≥ 0 with \|T(t)\| ≤ Meω t in a Banach space X and C is a linear operator in X such that D(A)⊂ D(C) and \| CR(μ ,A)\| ≤ K/(μ -ω) for each μ>ω, then, the above-mentioned evolution equation is well-posed, that is, $A+C$ generates a C₀-semigroup (TA+C(t) )t ≥ 0 satisfying \| TA+C(t)\| ≤ Me(ω +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.