Ill-posed problems associated with sectorial operators in Banach space

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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 [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<∞.

Original languageEnglish (US)
Article number124484
JournalJournal of Mathematical Analysis and Applications
Volume492
Issue number2
DOIs
StatePublished - Dec 15 2020

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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