Local and global solutions of well-posed integrated Cauchy problems

In this paper we study the local well-posed integrated Cauchy problem, v′(t) = Av(t) + tα Γ(α+ 1) x, v(0) = 0, t ∈ [0, κ), with κ> 0, α ≥ 0, and x ∈ X where X is a Banach space and A a closed operator on X. We extend solutions increasing the regularity in α. The global case (κ =∞) is also treated in detail. Growths of solutions are given in both cases. Key words: Abstract Cauchy problems, integrated semigroups, distribution semigroups