ABSTRACT
The methods devised for autonomous degenerate equations will be generalized for studying non autonomous degenerate equations. Precisely, in Section 4.1 we formulate a number of assumptions on A(t), similar to well known ones in semigroup theory, which allow to show that if f has the Hölder property (with exponent θ) and either f(0) + A(0)u 0 ∩ https://www.w3.org/1998/Math/MathML"> D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429165573/1ee49896-b2da-407e-820b-552c3e0f3cfb/content/eq1298.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ([−A(0)] θ ≠ ∅ or https://www.w3.org/1998/Math/MathML"> f ( 0 ) + A ( 0 ) u 0 ∩ X A ( 0 ) θ ≠ ∅ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429165573/1ee49896-b2da-407e-820b-552c3e0f3cfb/content/eq1299.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , then the Cauchy problem https://www.w3.org/1998/Math/MathML"> d u d t ∈ A ( t ) u + f ( t ) , 0 < t ≤ T , u ( 0 ) = u 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429165573/1ee49896-b2da-407e-820b-552c3e0f3cfb/content/eq1300.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> has precisely one strict solution (enjoying some further regularity, as well). Maximal regularity of the solutions is discussed in Section 4.2. Although in a first step we assume u 0 = 0 = f(0), we shall state the general result in Theorem 4.12 after a suitable reduction to this case.
