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Zbl 1081.34053
El-Borai, Mahmoud M.
The fundamental solutions for fractional evolution equations of parabolic type. (English)
[J] J. Appl. Math. Stochastic Anal. 2004, No. 3, 197-211 (2004). ISSN 1048-9533; ISSN 1687-2177/e

Let $0<\alpha\le 1$, $T> 0$, $E$ a Banach space, $\{A(t); t\in [0,T]\}$ a family of linear closed operators defined on dense set $D(A)$ in $E$ with values in $E$, $u: E\to E$ a function, $u_0\in D(A)$, $f: [0,T]\to E$ a given function and $B(E)$ be the Banach space of linear bounded operators $E\to E$ endowed with the topology defined by the operator norm. Assume that $D(A)$ is independent of $t$. Suppose that the operator $$(A(t)+\lambda I)^{-1}$$ exists in $B(E)$ for any $\lambda$ with $\text{Re\,}\lambda\ge 0$ and $$\align &(\exists C> 0)(\forall t\in [0,T])(\forall\lambda\in \bbfC)(\Vert(A(t)+\lambda I)^{-1}\Vert\le C(1+ |\lambda|)^{-1},\\ &(\exists C> 0)(\exists\gamma\in (0,1])(\forall(t_1, t_2,S)\in [0, T]^3)(\Vert(A(t_2)- A(t_1)(A^{-1}(s))\Vert\le C|t_2- t_1|^\gamma),\\ &(\exists C> 0)(\exists\beta\in (0,1])(\forall(t_1, t_2)\in [0,T]^2)(\Vert(f(t_2)- f(t_1)\Vert\le C|t_2- t_1|^\beta). \endalign$$ The author considers the fractional integral evolution equation $$u(t)= u_0- (\Gamma(\alpha))^{-1} \int^t_0 (t-\theta)^{\alpha-1} (A(\theta) u(\theta)- f(\theta))\,d\theta,\tag1$$ where $\Gamma: (0,+\infty)\to \bbfC$ is the Gamma-function, constructs the fundamental solution of the homogeneous fractional differential equation $${d^\alpha v(t)\over dt^\alpha}+ A(t) v(t)= 0,\qquad t> 0,\tag2$$ and proves the existence and uniqueness of the solution of (2) with the initial condition $v(0)= u_0$.\par 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.
[D. M. Bors (Iaşi)]

MSC 2000:: *34G10 Linear ODE in abstract spaces
35K99 Parabolic equations and systems
45J05 Integro-ordinary differential equations
26A33 Fractional derivatives and integrals (real functions)

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