Entire solutions of higher order abstract Cauchy problems (Q1359577)

The abstract Cauchy problem for the higher order differential equation \[ u^{(n)} (t)+ \sum^{n-1}_{i=0} A_ju^{(j)} (t)=0,\;t\geq 0;\;u^{(j)} (0)= u_j,\;0\leq j\leq n-1 \tag{1} \] is considered. The operators \(A_0\), \(A_1, \dots, A_{n-1}\) are closed linear operators with ranges and domains contained in a suitable complex Banach space \(E\). The concept of the entire solution of (1) is introduced and its existence and uniqueness is investigated. In particular, some conditions ensuring that (1) has a unique entire solution for every initial datum in a dense set are found. These conditions are satisfied for parabolic higher order linear differential equations.