arXiv subjects

Vu Trong Luong

Publications and source records attributed to Vu Trong Luong.

9 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 $\| T(t)\| \le Me^{ω_0 t}$, $t\ge 0$, 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 \[ \| C\|_A:= (1/M) \sup_{μ>ω_0 } \| (μ-ω_0) CR(μ,A)\| <\infty. \] 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 $B(\cdot)R(μ, A)$ as a function $[a,b]\to \mathcal{L}(\mathbb{X})$ is continuously differentiable, and \[ \limsup_{μ\to\infty} \sup_{t\in [a,b]} \left \| \frac{d}{dt}[B(t)R(μ,A)]\right \| <\infty. \] Examples are given to illustrate the obtained results.

math.DS↗

arXiv2025

Traveling waves in nonclassical diffusion equations

We study the existence of monotone traveling wave solutions in a class of nonclassical diffusion equations that include both standard diffusion and a higher-order mixed space-time dispersive term. The reaction term is nonlinear and subject to general structural conditions. By employing the method of upper and lower solutions, using less smooth super and subsolutions, we construct a monotone iterative scheme within a convex set and prove its convergence using Schauder's fixed point theorem. Explicit constructions of super and subsolutions are provided.

math.AP↗

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 $\|T(t)\| \le Me^{ωt}$ 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.

math.DS↗

arXiv2024

Existence of bounded asymptotic solutions of autonomous differential equations

We study the existence of bounded asymptotic mild solutions to evolution equations of the form $u'(t)=Au(t)+f(t), t\ge 0$ in a Banach space $\X$, where $A$ generates an (analytic) $C_0$-semigroup and $f$ is bounded. We find spectral conditions on $A$ and $f$ for the existence and uniqueness of asymptotic mild solutions with the same "profile" as that of $f$. In the resonance case, a sufficient condition of Massera type theorem is found for the existence of bounded solutions with the same profile as $f$. The obtained results are stated in terms of spectral properties of $A$ and $f$, and they are analogs of classical results of Katznelson-Tzafriri and Massera for the evolution equations on the half line. Applications from PDE are given.

math.DS↗

arXiv2023

On asymptotic periodic solutions of fractional differential equations and applications

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $ D^α_Cu(t)=Au(t)+f(t), u(0)=x, 0<α\le1, ( *) $ where $D^α_Cu(t)$ is the derivative of the function $u$ in the Caputo's sense, $A$ is a linear operator in a Banach space $\X$ that may be unbounded and $f$ satisfies the property that $\lim_{t\to \infty} (f(t+1)-f(t))=0$ which we will call asymptotic $1$-periodicity. By using the spectral theory of functions on the half line we derive analogs of Katznelson-Tzafriri and Massera Theorems. Namely, we give sufficient conditions in terms of spectral properties of the operator $A$ for all asymptotic mild solutions of Eq. (*) to be asymptotic $1$-periodic, or there exists an asymptotic mild solution that is asymptotic $1$-periodic.

math.CA↗

arXiv2020

A Spectral Theory of Polynomially Bounded Sequences and Applications to the Asymptotic Behavior of Discrete Systems

In this paper using a transform defined by the translation operator we introduce the concept of spectrum of sequences that are bounded by $n^ν$, where $ν$ is a natural number. We apply this spectral theory to study the asymptotic behavior of solutions of fractional difference equations of the form $Δ^αx(n)=Tx(n)+y(n)$, $n\in \mathbb{N}$, where $0<α\le 1$. One of the obtained results is an extension of a famous Katznelson-Tzafriri Theorem, saying that if the $α$-resolvent operator $S_α$ satisfies $\sup_{n\in\mathbb{N}} \| S_α(n)\| /n^ν<\infty$ and for all $z_0\in \{z\in \mathbb{C}: \ |z|=1\}$, but $z_0=1$, the complex function $(z^{1-α}(z-1)^α-T)^{-1}$ \ exists and is holomorphic in a neighborhood of $z_0$, then \begin{align*} \lim_{n\to \infty} \frac{1}{n^ν} \sum_{k=0}^{ν+1} \frac{(ν+1)!}{k!(ν+1-k)!} (-1)^{ν+1+k} S_α(n+k) =0. \end{align*} Three concrete examples are also included to illustrate the obtained results.

math.DS↗

arXiv2020

Asymptotic Behavior of Polynomially Bounded Solutions of Linear Fractional Differential Equations

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $D^α_Cu(t)=Au(t)+f(t)$ on the half line, where $D^α_Cu(t)$ is the derivative of the function $u$ in Caputo's sense, $A$ is generally an unbounded closed operator, $f$ is polynomially bounded. To this end we develop a spectral theory for functions of polynomial growth on the half line. Our main result claims that if $u$ is mild solution of the Cauchy problem such that $\lim_{h\downarrow 0} \sup_{t\ge 0} \| u(t+h)-u(t)\|/(1+t)^n=0$, and $\sup_{t\ge 0} \| u(t)\| /(1+t)^n <\infty$, then, $\lim_{t\to\infty} u(t)/(1+t)^n =0$ provided that the spectral set $Σ(A,α)\cap i\R$ is countable, where $Σ(A,α)$ is defined to be the set of complex numbers $ξ$ such that $λ^{α-1} (λ^α-A)^{-1}$ is analytic in a neighborhood of $ξ$, and $u$ satisfies some ergodic

math.DS↗

arXiv2018

Almost periodic solutions of periodic linear partial functional differential equations

We study conditions for the abstract periodic linear functional differential equation $\dot{x}=Ax+F(t)x_t+f(t)$ to have almost periodic with the same structure of frequencies as $f$. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the frequencies of the forcing term $f$. The obtained results extend recent results on the subject. A discussion on how the results could be extended to the case when $A$ depends on $t$ is given.

math.AP↗

arXiv2018

Asymptotic Behavior of Solutions of periodic linear partial functional differential equations on the half line

We study conditions for the abstract linear functional differential equation $\dot{x}=Ax+F(t)x_t+f(t), t\ge 0$ to have asymptotic almost periodic solutions, where $F(\cdot )$ is periodic, $f$ is asymptotic almost periodic. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the circular spectrum of the forcing term $f$. The obtained results extend recent results on the subject.

math.DS↗