A generation theorem for the perturbation of strongly continuous semigroups by unbounded operators
Xuan-Quang Bui, Nguyen Duc Huy, Vu Trong Luong, Nguyen Van Minh
TL;DR
The paper studies the well-posedness of the evolution equation $u'(t)=Au(t)+Cu(t)$ where $A$ generates a $C_0$-semigroup and $C$ may be unbounded. It employs the Hille–Yosida theorem and the resolvent identity $( u-(A+C))^{-1}=R( u,A)[I-CR( u,A)]^{-1}$ under the conditions $D(A) subseteq D(C)$ and $ orm{CR( u,A)}\le rac{K}{ u-oldsymbol{ u}}$ for all $ u>oldsymbol{ u}$ to show that $A+C$ generates a $C_0$-semigroup with explicit growth bounds $ orm{T_{A+C}(t)}\le M e^{(oldsymbol{ u}+MK)t}$. The work provides a new perturbation generation theorem for unbounded perturbations, introduces the perturbation space $\mathcal{GL}_A(\mathbb{X})$ and the Yosida distance $d_Y$, and uses these tools to analyze persistence of asymptotic behavior such as exponential dichotomy under small perturbations. These results extend classical perturbation theory for $C_0$-semigroups by giving concrete resolvent-based criteria and a framework for stability analysis under unbounded perturbations. The findings have implications for robustness of dynamical systems governed by linear evolution equations under perturbations.
Abstract
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 $\|T(t)\| \le Me^{ωt}$ in a Banach space $\mathbb{X}$ and $C$ is a linear operator in $\mathbb{X}$ such that $D(A)\subset D(C)$ and $\| CR(μ,A)\| \le K/(μ-ω)$ for each $μ>ω$, 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 $\| T_{A+C}(t)\| \le Me^{(ω+MK)t}$. 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.