On the solution theory of factored Cauchy problems and the abstract d'Alembert's formula (Q1583904)

The paper deals with the mild solutions of \(n-\)th order factored (or iterated) abstract Cauchy problems in the following sense. Considering on the Banach space \((X,\|\cdot\|)\) a sequence \(A_1,\dots,A_n\) of infinitesimal generator of the semigroups \((T_1(t))_{t\geq 0},\dots, (T_n(t))_{t\geq 0}\) respectively. A mild solutions of \(n\)th order factored abstract Cauchy problems is a solution of the following Cauchy problem \[ \biggr({d\over dt}-A_1\biggr)\cdots\biggr({d\over dt}-A_n\biggr)u=f\in C([0,\infty);C), \quad {d^{j-1}u\over dt^{(j-1)}}(0)=x_j,\;j=1,\dots,n. \tag{CP} \] Under some assumptions, the author show that the cauchy problem (CP) is well posed. Some particular results also are given for some additional assumptions such that the commutativity of the family \(A_1,\cdots,A_n\). At the end of this work, some examples on Klein-Gordon equations, Damped wave equations, Linear elasticity equations and more are given.