Regularized semigroups, iterated Cauchy problems and equipartition of energy (Q1261142)

Among all higher order linear differential equations \(u^{(n)}(t)+ A_{n-1} u^{(n-1)}(t)+ \cdots +A_ 1 u'(t)+A_ 0 u(t)=0\) in Banach spaces, \textit{J. Sandefur} [J. Math. Anal. Appl. 60, 728-742 (1977; Zbl 0358.35068)] has singled out those that can be written in the form (1) \(\prod_{k=1}^ n (D-A_ k)u(t)=0\), where \(D=d/dt\) and each \(A_ k\) is a semigroup generator. Recently, the class of regularized semigroups that extend strongly continuous semigroups has been introduced; equations have been pointed out that can be treated within this extended class but not by means of strongly continuous semigroup theory. The authors study equation (1) with each \(A_ k\) the generator of a regularized semigroup and obtain results dealing with well posedness of the initial value problem and equipartition of energy of solutions for certain equations.