INCOMPLETE ITERATED CAUCHY-PROBLEMS

We treat numerous higher order incomplete abstract Cauchy problems by writing them in the form 1. (1) Πk = 1n, ( d dt - Ak) u(t) = 0 (t ≥ 0), 2. (2) ui - 1(0) = xi (1 ≤ i ≤ j), 3. (3a) ui - 1(t) bounded (1 ≤ i ≤ n - j), or 4. (3b) limt → ∞ u(i - 1)(t) = 0 (1 ≤ i ≤ n - j). The abstract Cauchy problem (1), (2), and (3a) is wellposed, in a certain sense, when A1, ..., Aj generate bounded C-semigroups and (-Aj + 1), ..., (-An) generate uniformly stable C-semigroups. For (1), (2), and (3b), we need A1, ..., Aj and (-Aj + 1), ..., (-An) to generate stable C-semigroups. © 1992.