# Article

 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 . Category: math . 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 . 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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