Ill-posed problems associated with sectorial operators in Banach space

Abstract

Ill-posed problems of the form u′(t)=Au+h, 0<t<T, u(0)=φ are studied in Banach space under two related scenarios, both of which consider a well-posed logarithmic approximation. In the first scenario, −A generates a bounded holomorphic semigroup of angle θ=π2, and so A is sectorial of angle 0. In the second, where θ may be relaxed to any value in (0,π2], we impose an additional assumption that e−A is sectorial. Regularization results for the original ill-posed problem are then established in the linear case where h=h(t), and subsequently extended to nonlinear problems where h is replaced by h(t,u(t)). We compare the two cases in terms of regularization convergence rates and apply our results to backward heat equations, both linear and nonlinear in Lp(Rn), 1<p<∞.