Regularization for Ill-posed Cauchy problems associated with generators of analytic semigroups

Abstract

This paper is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. Our main result is that if − A is the generator of an analytic semigroup, then there exists a family of regularizing operators for such an ill-posed Cauchy problem by using the quasi-reversibility method, fractional powers and semigroups of linear operators. The applications to ill-posed partial differential equations are also given.