Ill-posed problems associated with sectorial operators in Banach space

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 [Formula presented], and so A is sectorial of angle 0. In the second, where θ may be relaxed to any value in [Formula presented], 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<∞.

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Work Title Ill-posed problems associated with sectorial operators in Banach space
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Open Access
Creators
  1. Matthew A. Fury
Keyword
  1. Ill-posed problem
  2. Backward heat equation
  3. Semigroup of linear operators
  4. Sectorial operator
License CC BY-NC-ND 4.0 (Attribution-NonCommercial-NoDerivatives)
Work Type Article
Publisher
  1. Journal of Mathematical Analysis and Applications
Publication Date August 10, 2020
Publisher Identifier (DOI)
  1. https://doi.org/10.1016/j.jmaa.2020.124484
Deposited June 20, 2023

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    Keyword
    • ill-posed problem, Backward heat equation, Semigroup of linear operators, Sectorial operator
    Publication Date
    • 2020-12-15
    • 2020-08-10
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    Keyword
    • ill-posed problem, Backward heat equation, Semigroup of linear operators, Sectorial operator
    • Ill-posed problem, Backward heat equation, Semigroup of linear operators, Sectorial operator
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