Well-posedness of the time-dependent linear Cauchy problem

Abstract

In the second chapter we will study well-posedness of the linear, time-dependent Cauchy problem

$$ \frac{{du}}{{dt}}(t) = A(t)u(t),u({t_0}) = {u_0}$$

for operators A(t) ∈ ∩_{k∈IN 0} ℒ(X^k+m, X^k) of order m in a scale of Banach spaces (X^k) _k (i.e., X^k for k ∈ ℕ₀ is a Banach space with X^k ↪ X^l for k ≥ l). Here well-posedness roughly means that for sufficiently smooth initial values u₀ there are unique solutions depending continuously on u₀.