Asymptotic behavior of fractional order semilinear evolution equations

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.