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) = chi where - A generates a holomorphic semigroup of angle 0 is an element of (0, pi/2] on a Banach space X and a is an element of C([0, T]: R+). The problem is generally ill-posed under such conditions, and so we employ methods to approximate known solutions of the problem. 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 theta is an element of (0, pi/4] or theta is an element of (pi/4, pi/2], we provide two separate approximations each yielding a regularizing family. The theory has applications to ill-posed partial differential equations in L-p(Omega), 1 < p < infinity where A is a strongly elliptic differential operator and Omega is a fixed domain in R-n.