Uniqueness for an inverse problem for a nonlinear parabolic system with an integral term by one-point Dirichlet data
type: Item
instance of: scholarly article
title: Uniqueness for an inverse problem for a nonlinear parabolic system with an integral term by one-point Dirichlet data
zbMATH Open document ID: 1417.80002
author: Dietmar Hömberg
author: Shuai Lu
author: M. Yamamoto
published in: Journal of Differential Equations
publication date: 2019-03-22T00:00:00Z
review text: Laser-induced thermotherapy (LITT) is a cancer treatment that requires minimal invasion. The mathematical model of the LITT includes the so-called bio-heat equation for the unknown temperature $u$ in the tumor tissue and the Arrhenius-type equation for the distribution of native tissues. The key term of the model is the temperature-dependent function $f(u)$ that describes the coagulation growth kinetics. \par One can measure the temperature at the only point at the tumor tissue's boundary, and the inverse problem arises from the determination of $f$ based on this one-point measurement. Before beginning to solve the problem it is important to know if its solution is unique. To prove this fact, the authors first formulate and prove a comparison principle for temperatures $u$ in the LITT model that corresponds to different functions $f$. The uniqueness of the classical solution for the direct problem (with known $f$) is also discussed.
reviewed by: Aleksey Syromyasov
Mathematics Subject Classification ID: 80A23
Mathematics Subject Classification ID: 92C50
Mathematics Subject Classification ID: 78A70
Mathematics Subject Classification ID: 35K10
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35A02
Mathematics Subject Classification ID: 92C37
Mathematics Subject Classification ID: 78A60
zbMATH DE Number: 7040866
zbMATH Keywords: bio-heat equation
zbMATH Keywords: inverse problem
zbMATH Keywords: comparison principle
zbMATH Keywords: solution uniqueness
zbMATH Keywords: laser-induced thermotherapy
MaRDI profile type: Publication
full work available at URL: j.jde.2018.12.004
OpenAlex ID: W2903733062
cites work: On a mathematical model for laser-induced thermotherapy
cites work: Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition
cites work: Unique Identification of Eigenvalues and Coefficients in a Parabolic Problem
cites work: Uniqueness from pointwise observations in a multi-parameter inverse problem
cites work: Stable determination of a semilinear term in a parabolic equation
cites work: Unicity in an inverse problem for an unknown reaction term in a reaction- diffusion equation
cites work: Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation
cites work: Global uniqueness of a multidimensional inverse problem for a nonlinear parabolic equation by a Carleman estimate
cites work: An inverse problem for a semilinear parabolic equation
cites work: An Inverse problem for a nonlinear parabolic equation
cites work: Identification of the thermal growth characteristics of coagulated tumor tissue in laser-induced thermotherapy
cites work: Identification of nonlinear heat transfer laws by optimal control
cites work: Parameter identification in non-isothermal nucleation and growth processes
cites work: Geometric theory of semilinear parabolic equations
cites work: Q5286659
cites work: Q5562267
cites work: Semigroups of linear operators and applications to partial differential equations
Wikidata QID: Q128832825
DOI: 10.1016/J.JDE.2018.12.004
description: scientific article; zbMATH DE number 7040866