On regularization and error estimates for non-homogeneous backward Cauchy problem

Abstract

In this paper we study the general non-homogeneous Backward Cauchy problemut+Au=f,0<t<T,u(T)=g,for positive selfadjoint unbounded operator A on the Hilbert space H. The problem is known to be severely ill-posed. We give extensions of the quasi-boundary methods to the non-homogeneous case. We prove several sharp results on regularizations and error-estimates. Other results, including some explicit convergence rates are proved. Finally applications to the non-homogeneous backward heat equation with Bessel operator are given.