Asymptotic behavior of fractional order semilinear evolution equations. (English) Zbl 1299.35309

Summary: 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 \(S_{\theta }:=\{\lambda \in \mathbb {C}\setminus \{0\}\: | \arg (\lambda )| <\theta +\pi /2\}\subset \rho (A)\) (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 \[ D_t^{\alpha +1}u(t)+\mu D_t^{\beta }u(t)=Au(t)+\frac {t^{-\alpha }}{\Gamma (1-\alpha )}u'(0)+\mu \frac {t^{-\beta }}{\Gamma (1-\beta )}u(0)+f(t,u(t)),\; t>0, \] \(0<\alpha \leq \beta \leq 1\), \(\mu >0\) 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 \(1/\Gamma (0)=0.\) The general result on the asymptotic behavior is obtained by first establishing a sharp estimate on the solution family associated to the linear equation.


35R11 Fractional partial differential equations
47D06 One-parameter semigroups and linear evolution equations
35B40 Asymptotic behavior of solutions to PDEs