## 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.

### MSC:

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