Differential and Integral Equations

Asymptotic behavior of fractional order semilinear evolution equations

Valentin Keyantuo, Carlos Lizama, and Mahamadi Warma

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Fractional calculus is a subject of great interest in many areas of mathematics, physics, and sciences, including stochastic processes, mechanics, chemistry, and biology. We will call an operator $A$ on a Banach space $X$ $\omega$-sectorial ($\omega\in\mathbb R$) of angle $\theta$ if there exists $\theta \in [0,\pi/2)$ such that $S_\theta:=\{\lambda\in\mathbb C\setminus\{0\} : |\mbox{arg} (\lambda)| < \theta+\pi/2\}\subset\rho(A)$ (the resolvent set of $A$) and $\sup\{|\lambda-\omega|\|(\lambda-A)^{-1}\| : \lambda\in \omega +S_\theta\} < \infty$. Let $A$ be $\omega$-sectorial of angle $\beta\pi/2$ with $\omega < 0$ and $f$ an $X$-valued function. Using the theory of regularized families, and Banach's fixed-point theorem, we prove existence and uniqueness of mild solutions for the semilinear fractional-order differential equation \begin{align*} & D_t^{\alpha+1}u(t) + \mu D^{\beta}_t u(t) \\ & = Au(t) + \frac{t^{-\alpha}}{\Gamma(1-\alpha)}u'(0) + \mu \frac{t^{-\beta}}{\Gamma(1-\beta)} u(0) + f(t,u(t)), \,\, t > 0, \end{align*} $0 < \alpha \leq \beta \leq 1,\,\, \mu >0$, with the property that the solution decomposes, uniquely, into a periodic term (respectively almost periodic, almost automorphic, compact almost automorphic) and a second term that decays to zero. We shall make the convention $\frac{1}{\Gamma(0)}=0.$ The general result on the asymptotic behavior is obtained by first establishing a sharp estimate on the solution family associated to the linear equation.

Article information

Source
Differential Integral Equations, Volume 26, Number 7/8 (2013), 757-780.

Dates
First available in Project Euclid: 20 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1369057816

Mathematical Reviews number (MathSciNet)
MR3098986

Zentralblatt MATH identifier
1299.35309

Subjects
Primary: 34A08: Fractional differential equations 35R11: Fractional partial differential equations 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 45N05: Abstract integral equations, integral equations in abstract spaces

Citation

Keyantuo, Valentin; Lizama, Carlos; Warma, Mahamadi. Asymptotic behavior of fractional order semilinear evolution equations. Differential Integral Equations 26 (2013), no. 7/8, 757--780. https://projecteuclid.org/euclid.die/1369057816


Export citation