arXiv subjects

Xuan-Quang Bui

Publications and source records attributed to Xuan-Quang Bui.

4 recordsLinked to original sources

arXiv2026

On the well-posedness of linear evolution equations under unbounded nonautonomous perturbations

We study conditions for the well-posedness of nonautonomous perturbation of evolution equations of the form \[ u'(t)=(A+B(t))u(t), \quad t \in [a,b], \] where $A$ generates a $\mathrm{C}_0$-semigroup $\left (T(t)\right )_{t\ge 0}$ with $\| T(t)\| \le Me^{\omega_0 t}$, $t\ge 0$, in a Banach space $\mathbb{X}$ and $B(t)$ are $t$-dependent (unbounded) linear operators in $\mathbb{X}$. The unbounded perturbation operators $B(t)$ are assumed to belong to a normed space (denoted by $\mathcal{GL}_A (\mathbb{X})$) of unbounded linear operators $C$ in $\mathbb{X}$ such that $D(A) \subset D(C)$ with norm \[ \| C\|_A:= (1/M) \sup_{\mu >\omega_0 } \| (\mu-\omega_0) CR(\mu,A)\| <\infty. \] We prove that the above-mentioned evolution equation admits an evolution family if $\| B(\cdot)\|_A$ is continuous in $[a,b]$. The evolution family is unique if $B(\cdot)R(\mu, A)$ as a function $[a,b]\to \mathcal{L}(\mathbb{X})$ is continuously differentiable, and \[ \limsup_{\mu \to\infty} \sup_{t\in [a,b]} \left \| \frac{d}{dt}[B(t)R(\mu,A)]\right \| <\infty. \] Examples are given to illustrate the obtained results.

math.DS↗

arXiv2024

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.

math.DS↗

arXiv2023

Roughness of exponential dichotomy under unbounded perturbation in linear partial functional differential equations

This paper is concerned with the roughness of exponential dichotomies under unbounded perturbations of a class of linear partial functional differential equations \begin{equation}\label{pfde-000-1star} u'(t)=Au(t)+Bu_t, \end{equation} where $A$ is a linear operator on a Banach space $\mathbb{X}$ and $B$ is a linear operator from $C([-r,0],\mathbb{X})$ into $\mathbb{X}$, where $r>0$ is a given constant. To quantify the size of unbounded perturbations, we introduce the \textit{Yosida distance} between linear operators $U$ and $V$, defined by $d_Y(U,V):=\limsup_{\mu\to +\infty} \| U_\mu-V_\mu\|$, where $U_\mu$ and $V_\mu$ are the Yosida approximations of $U$ and $V$, respectively. We show that if $d_Y(A, A_1)$ and $d_Y(B, B_1)$ are sufficiently small, then the perturbed equation \begin{equation}\label{pfde-000-2star} u'(t)=A_1u(t)+B_1u_t \end{equation} also admits an exponential dichotomy whenever \eqref{pfde-000-1star} admits one. The proofs are based on estimates of the Yosida distance between the generators of the solution semigroups associated with \eqref{pfde-000-1star} and \eqref{pfde-000-2star} in the phase space $C([-r,0],\mathbb{X})$, without assuming any relation between their domains.

math.DS↗

arXiv2023

Yosida Distance and Existence of Invariant Manifolds in the Infinite-Dimensional Dynamical Systems

We introduce a new concept of Yosida distance between two (unbounded) linear operators $A$ and $B$ in a Banach space $\mathbb{X}$ defined as $d_Y(A,B):=\limsup_{μ\to +\infty} \| A_μ-B_μ\|$, where $A_μ$ and $B_μ$ are the Yosida approximations of $A$ and $B$, respectively, and then study the persistence of evolution equations under small Yosida perturbation. This new concept of distance is also used to define the continuity of the proto-derivative of the operator $F$ in the equation $u'(t)=Fu(t)$, where $F \colon D(F)\subset \mathbb{X} \rightarrow \mathbb{X}$ is a nonlinear operator. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of $F$ is continuous. The Yosida distance approach to perturbation theory allows us to free the requirement on the domains of the perturbation operators. Finally, the obtained results seem to be new.

math.DS↗