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Title: Truncated spectral regularization for an ill-posed non-linear parabolic problem (English)
Author: Jana, Ajoy
Author: Nair, M. Thamban
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 69
Issue: 2
Year: 2019
Pages: 545-569
Summary lang: English
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Category: math
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Summary: It is known that the nonlinear nonhomogeneous backward Cauchy problem $u_t(t)+Au(t)=f(t,u(t))$, $0\leq t<\tau $ with $u(\tau )=\phi $, where $A$ is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on $\phi $ and $f$, that a solution of the above problem satisfies an integral equation involving the spectral representation of $A$, which is also ill-posed. Spectral truncation is used to obtain regularized approximations for the solution of the integral equation, and error analysis is carried out with exact and noisy final value $\phi $. Also stability estimates are derived under appropriate parameter choice strategies. This work extends and generalizes many of the results available in the literature, including the work by Tuan (2010) for linear homogeneous final value problem and the work by Jana and Nair (2016b) for linear nonhomogeneous final value problem. (English)
Keyword: ill-posed problem
Keyword: nonlinear parabolic equation
Keyword: regularization
Keyword: parameter choice
Keyword: semigroup
Keyword: contraction principle
MSC: 35K55
MSC: 35R30
MSC: 47A52
MSC: 47H10
MSC: 65F22
MSC: 65M12
idZBL: Zbl 07088804
idMR: MR3959964
DOI: 10.21136/CMJ.2019.0435-17
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Date available: 2019-05-24T09:02:41Z
Last updated: 2019-09-26
Stable URL: http://hdl.handle.net/10338.dmlcz/147744
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