Cauchy problem for abstract evolution equations of parabolic type (Q2276667)

The fundamental solution U(t,s) of the abstract evolution equation \[ (1)\quad du/dt+A(t)u=f(t),\quad u(0)=u_ 0 \] of parabolic type in a Banach space E is constructed under a very mild assumption on the smoothness of A(t) in t. It is assumed that the domain of A(t) does not depend on t and the following inequality holds \[ \| (A(t)- A(\tau))A(s)^{-1}\| \leq \omega (| t-\tau |), \] where \(\omega\) (r) is a positive increasing function such that \[ \int^{T_ 0}_{0}\omega (r)| \log r| r^{-1}dr<\infty,\quad \omega (r)| \log r| \to 0\quad as\quad r\downarrow 0. \] In the construction of the fundamental solution the author follows the method of \textit{P. E. Sobolevskij} [Tr. Mosk. Mat. O.-va 10, 297-350 (1961; Zbl 0141.327)], where \(\omega (r)=r^{\alpha}\) \((0<\alpha <1)\). With the aid of the fundamental solution a unique solution of (1) is expressed by Duhamel's principle for any \(u_ 0\in E\) and f satisfying \[ \int^{t}_{\tau}(\| f(t)-f(s)\|)/(t-s)ds\to 0\quad as\quad t- \tau \downarrow 0. \]