Nonautonomous ill-posed evolution problems with strongly elliptic differential operators

In this article, we consider the nonautonomous evolution problem du/dt= a(t)Au(t); 0≤ s ≤t < T with initial condition u(s)= X where -A generates a holomorphic semigroup of angle θ{cyrillic, ukrainian} (0, π/2] on a Banach space X and {cyrillic, ukrainian} C([0, T]: ℝ⁺). The problem is generally ill-posed under such condi- tions, and so we employ methods to approximate known solutions of the prob- lem. In particular, we prove the existence of a family of regularizing operators for the problem which stems from the solution of an approximate well-posed problem. In fact, depending on whether θ{cyrillic, ukrainian} (0, π/4] or θ{cyrillic, ukrainian} (π/4, π/2], we provide two separate approximations each yielding a regularizing family. The theory has applications to ill-posed partial difierential equations in L^p(Ω), 1< p< ∞ where A is a strongly elliptic difierential operator and Ω fixed domain in ℝⁿ.