A generation theorem for the perturbation of strongly continuous semigroups by unbounded operators
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arXiv
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| Format: | Preprint |
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2024
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| author | Bui, Xuan-Quang Huy, Nguyen Duc Luong, Vu Trong Van Minh, Nguyen |
| author_facet | Bui, Xuan-Quang Huy, Nguyen Duc Luong, Vu Trong Van Minh, Nguyen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06812 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A generation theorem for the perturbation of strongly continuous semigroups by unbounded operators Bui, Xuan-Quang Huy, Nguyen Duc Luong, Vu Trong Van Minh, Nguyen Dynamical Systems Functional Analysis 34G10, 47D06 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. |
| title | A generation theorem for the perturbation of strongly continuous semigroups by unbounded operators |
| topic | Dynamical Systems Functional Analysis 34G10, 47D06 |
| url | https://arxiv.org/abs/2405.06812 |