Ill-posed problems associated with sectorial operators in Banach space

[摘要] Ill-posed problems of the form u' (t) = Au + h, 0 < t < T, u(0) = phi are studied in Banach space under two related scenarios, both of which consider a wellposed logarithmic approximation. In the first scenario, -A generates a bounded holomorphic semigroup of angle theta = pi/2, and so A is sectorial of angle 0. In the second, where theta may be relaxed to any value in (0, pi/2], 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 his 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 L-p(R-n), 1 < p < infinity. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-12-15

[关键词] Ill-posed problem;Backward heat equation;Semigroup of linear operators;Sectorial operator