The fundamental solutions for fractional evolution equations of parabolic type
Let \(0<\alpha\leq 1\), \(T> 0\), \(E\) 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 \(\text{Re\,}\lambda\geq 0\) and \[ \begin{aligned} &(\exists C> 0)(\forall t\in [0,T])(\forall\lambda\in \mathbb{C})(\|(A(t)+\lambda I)^{-1}\|\leq C(1+ |\lambda|)^{-1},\\ &(\exists C> 0)(\exists\gamma\in (0,1])(\forall(t_1, t_2,S)\in [0, T]^3)(\|(A(t_2)- A(t_1)(A^{-1}(s))\|\leq C|t_2- t_1|^\gamma),\\ &(\exists C> 0)(\exists\beta\in (0,1])(\forall(t_1, t_2)\in [0,T]^2)(\|(f(t_2)- f(t_1)\|\leq C|t_2- t_1|^\beta). \end{aligned} \] The author considers the fractional integral evolution equation \[ u(t)= u_0- (\Gamma(\alpha))^{-1} \int^t_0 (t-\theta)^{\alpha-1} (A(\theta) u(\theta)- f(\theta))\,d\theta,\tag{1} \] where \(\Gamma: (0,+\infty)\to \mathbb{C}\) is the Gamma-function, constructs the fundamental solution of the homogeneous fractional differential equation \[ {d^\alpha v(t)\over dt^\alpha}+ A(t) v(t)= 0,\qquad t> 0,\tag{2} \] and proves the existence and uniqueness of the solution of (2) with the initial condition \(v(0)= u_0\). 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.
scientific article; zbMATH DE number 2148074