I. SEMI LINEAR EQUATIONS OF EVOLUTION

et X be a Banach space. A one parameter family S(t) (t ≥ 0) of bounded linear operators on X is a strongly continuous semigroup, ( C0 is semigroup for short) if (a) S(0) = I which I is the identity operator on X. (b) S(t + s) = S(t)S(s) for every t, s> 0. (c) lim t→0 S(t)x = x for every x ∈ X (strong continuity). As a quite direct application of the uniform boundedness theorem, there exist ω ≥ 0 and M ≥ 1 such that ⃦S(t) ⃦ L(X) ≤ M, ∀ t ≥ 0. This in turn entails the continuity of the map t S(t)x from [0,∞) to X, for every fixed x ∈ X.The linear operator A domai