Fundamental solution with weak singularity (Q2725209)

The author considers the abstract second-order linear initial value problem NEWLINE\[NEWLINE{d^2u\over dt^2}= B^2(t)u,\quad t\in (0,T],\quad u(0)= u_0,\quad {du\over dt}(0)= u_1,NEWLINE\]NEWLINE where \(X\) is a real Banach space, \(u_0,u_1\in X\), and \(\{B(t)\}_{t\in [0,T]}\) is a family of linear and closed operators on \(X\) under the following assumptions:NEWLINENEWLINENEWLINEi) \(D(B(t))\) and \(D(B^2(t))\) are each independent of \(t\) and \(\overline{D(B(t))}= X\),NEWLINENEWLINENEWLINEii) the resolvent set \(\rho(B(t))= \mathbb{C}\setminus\{0\}\),NEWLINENEWLINENEWLINEiii) there exist \(M> 0\) and \(\theta\in (2/3,1)\) such that \(\|R(\lambda; B(t))\|= \|(\lambda- B(t))^{-1}\|\leq {M\over |\lambda|^\theta}\), for \(\lambda\neq 0\), \(t\in [0,T]\),NEWLINENEWLINENEWLINEiv) for \(\lambda_0\neq 0\) there exist \(L>0\) and \(\alpha\in (0,1)\) such thatNEWLINENEWLINENEWLINE\(\|[B^2(t_1)- B^2(t_2)] R(\lambda_0; B^2(t))\|\leq L|t_1- t_2|^\alpha\) for \(t,t_1,t_2\in [0,T]\).NEWLINENEWLINENEWLINEUnder these conditions, the fundamental solution with weak singularity is constructed. The construction is accomplished by reducing the problem to a certain first-order initial value problem. This builds on results due to \textit{H. Tanabe} [Equations of evolution. Translated from Japanese by N. Mugibayashi and H. Haneda. (English) Monographs and Studies in Mathematics. 6. London-San Francisco-Melbourne: Pitman (1979; Zbl 0417.35003)], \textit{M. Kozak} [Zesz. Nauk. Uniw. Jagiell., Univ. Iagell. Acta Math. 1169(32), 275-289 (1995; Zbl 0855.34073)], and \textit{W. Obloza} [Bull. Pol. Acad. Sci., Math. 45, No. 3, 269-279 (1997; Zbl 0886.34054)].