A degenerate reaction diffusion system modelling an optical biosensor

Merino, Sandro; Grinfeld, Michael; McKee, Sean

doi:10.1007/s000330050082

Cited by 8 publications

(3 citation statements)

References 17 publications

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“…Such situations have been reported in several works, such as those modelling optical sensors or biosensors (see e.g. [BIK09,JJMS96,MGM98]). However, one can intuitively guess that due to the interactions between the concentration, governed by the chemical reactions, an effect of indirect diffusion will be observed, i.e.…”

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confidence: 78%

Indirect diffusion effect in degenerate reaction-diffusion systems

Einav¹,

Morgan²,

Tang³

2020

Preprint

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In this work we study global well-posedness and large time behaviour for a typical reaction-diffusion system, whose non-linearities arise from chemical reactions, and which include degenerate diffusion. We show that there is an indirect diffusion effect, i.e. an effective diffusion for the non-diffusive species which is incurred by a combination of diffusion from diffusive species and reversible reactions between the species. Deriving new estimates for such degenerate reaction-diffusion system, we show, by applying the entropy method, that the solution converges exponentially to equilibrium, and provide explicit convergence rates and associated constants. Contents 1. Introduction 1 1.1. Reaction-diffusion equations 1 1.2. The setting of the problem 2 1.3. The current state of the art 4 1.4. Main results and key ideas 5 2. Indirect diffusion effect and the entropy method 9 2.1. The entropic inequalities 9 2.2. Conditional convergence to equilibrium 18 3. Global existence of solutions 20 3.1. Auxiliary results 21 3.2. Proof of Proposition 2 24 3.3. Proof of Proposition 1 37 4. The interaction between the entropy and the norm bounds: Proof of the main theorems 38 5. Final Remarks 43 References 43

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confidence: 78%

Indirect diffusion effect in degenerate reaction-diffusion systems

Einav¹,

Morgan²,

Tang³

2020

Preprint

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“…1,3,[10][11][12]18 The main scheme of proving global existence for parabolic problems with nonlinear boundary conditions follows; 1 see also Ref. 17 for an accessible introduction to Amann's theory.…”

Section: Well-posedness and Asymptotic Behaviormentioning

confidence: 99%

A Mathematical Study of a Bistable Nematic Liquid Crystal Device

Costa

Grinfeld

Mottram

et al. 2007

Math. Models Methods Appl. Sci.

Self Cite

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We consider a model of a bistable nematic liquid crystal device based on the EricksenLeslie theory. The resulting mathematical object is a parabolic PDE with nonlinear dynamic boundary conditions. We analyze well-posedness of the problem and global existence of solutions using the theory developed by Amann. Furthermore, using phaseplane methods, we give an exhaustive description of the steady state solutions and hence of the switching capabilities of the device.

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“…A comprehensive study of the mathematical modelling of amperometric biosensors is given in [31]. Mathematical modelling in the design of optical biosensors has been applied in individual cases only [32, 33]. …”

Section: Introductionmentioning

confidence: 99%

Modelling a Peroxidase-based Optical Biosensor

Baronas

̇e

Kulys

2007

Sensors

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The response of a peroxidase-based optical biosensor was modelled digitally. A mathematical model of the optical biosensor is based on a system of non-linear reaction-diffusion equations. The modelling biosensor comprises two compartments, an enzyme layer and an outer diffusion layer. The digital simulation was carried out using finite difference technique. The influence of the substrate concentration as well as of the thickness of both the enzyme and diffusion layers on the biosensor response was investigated. Calculations showed complex kinetics of the biosensor response, especially at low concentrations of the peroxidase and of the hydrogen peroxide.

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A degenerate reaction diffusion system modelling an optical biosensor

Cited by 8 publications

References 17 publications

Indirect diffusion effect in degenerate reaction-diffusion systems

Indirect diffusion effect in degenerate reaction-diffusion systems

A Mathematical Study of a Bistable Nematic Liquid Crystal Device

Modelling a Peroxidase-based Optical Biosensor

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