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The fundamental solutions for fractional evolution equations of parabolic type. (English) Zbl 1081.34053

Let $0<\alpha \le 1$, $T>0$, $E$ a Banach space, $\left\{A\left(t\right);t\in \left[0,T\right]\right\}$ a family of linear closed operators defined on dense set $D\left(A\right)$ in $E$ with values in $E$, $u:E\to E$ a function, ${u}_{0}\in D\left(A\right)$, $f:\left[0,T\right]\to E$ a given function and $B\left(E\right)$ be the Banach space of linear bounded operators $E\to E$ endowed with the topology defined by the operator norm. Assume that $D\left(A\right)$ is independent of $t$. Suppose that the operator

${\left(A\left(t\right)+\lambda I\right)}^{-1}$

exists in $B\left(E\right)$ for any $\lambda$ with $\text{Re}\phantom{\rule{0.166667em}{0ex}}\lambda \ge 0$ and

$\begin{array}{cc}& {\left(\exists C>0\right)\left(\forall t\in \left[0,T\right]\right)\left(\forall \lambda \in ℂ\right)\left(\parallel \left(A\left(t\right)+\lambda I\right)}^{-1}{\parallel \le C\left(1+|\lambda |\right)}^{-1},\hfill \\ & \left(\exists C>0\right)\left(\exists \gamma \in \left(0,1\right]\right)\left(\forall \left({t}_{1},{t}_{2},S\right)\in {\left[0,T\right]}^{3}\right)\left(\parallel \left(A\left({t}_{2}\right)-A\left({t}_{1}\right)\left({A}^{-1}\left(s\right)\right)\parallel \le C|{t}_{2}-{t}_{1}{|}^{\gamma }\right),\hfill \\ & \left(\exists C>0\right)\left(\exists \beta \in \left(0,1\right]\right)\left(\forall \left({t}_{1},{t}_{2}\right)\in {\left[0,T\right]}^{2}\right)\left(\parallel \left(f\left({t}_{2}\right)-f\left({t}_{1}\right)\parallel \le C|{t}_{2}-{t}_{1}{|}^{\beta }\right)·\hfill \end{array}$

The author considers the fractional integral evolution equation

$u\left(t\right)={u}_{0}-{\left({\Gamma }\left(\alpha \right)\right)}^{-1}{\int }_{0}^{t}{\left(t-\theta \right)}^{\alpha -1}\left(A\left(\theta \right)u\left(\theta \right)-f\left(\theta \right)\right)\phantom{\rule{0.166667em}{0ex}}d\theta ,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where ${\Gamma }:\left(0,+\infty \right)\to ℂ$ is the Gamma-function, constructs the fundamental solution of the homogeneous fractional differential equation

$\frac{{d}^{\alpha }v\left(t\right)}{d{t}^{\alpha }}+A\left(t\right)v\left(t\right)=0,\phantom{\rule{2.em}{0ex}}t>0,\phantom{\rule{2.em}{0ex}}\left(2\right)$

and proves the existence and uniqueness of the solution of (2) with the initial condition $v\left(0\right)={u}_{0}$.

The author also proves the continuous dependence of the solutions of equation (1) on the element ${u}_{0}$ and the function $f$ and gives an application to a mixed problem of a parabolic partial differential equation of fractional order.

##### MSC:
 34G10 Linear ODE in abstract spaces 35K99 Parabolic equations and systems 45J05 Integro-ordinary differential equations 26A33 Fractional derivatives and integrals (real functions)