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Ohmic Heating Phenomena

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Non-Local Partial Differential Equations for Engineering and Biology

Part of the book series: Mathematics for Industry ((MFI,volume 31))

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Abstract

The current chapter considers two main applications associated with Ohmic heating phenomena. Initially we deal with an application from food industry, building up two one-dimensional non-local problems illustrating the evolution of the temperature of the sterilized food. The former model consists of a diffusion-convection equation while the latter of a convection equation with non-local convection velocity. Both of these non-local models are investigated in terms of their stability and the occurrence of finite-time blow-up, where the latter in the current context indicates food burning. Different approaches should be followed though depending on the monotonicity of the nonlinearity appearing in the non-local term, since no maximum principle is available for the non-local parabolic problem when this nonlinearity is increasing. The second part of the chapter is devoted to the study of a non-local parabolic model illustrating the operation of the thermistor device. Notably, conditions under which finite-time blow-up, which here indicates the destruction of the thermistor device, occurs are investigated by using both energy and comparison methods.

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Authors and Affiliations

  1. Department of Mathematics, University of Chester, Chester, UK

    Nikos I. Kavallaris

  2. Department of Mathematics, Osaka University, Osaka, Japan

    Takashi Suzuki

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  1. Nikos I. Kavallaris
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  2. Takashi Suzuki
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Correspondence to Nikos I. Kavallaris .

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Kavallaris, N.I., Suzuki, T. (2018). Ohmic Heating Phenomena. In: Non-Local Partial Differential Equations for Engineering and Biology. Mathematics for Industry, vol 31. Springer, Cham. https://doi.org/10.1007/978-3-319-67944-0_2

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  • DOI: https://doi.org/10.1007/978-3-319-67944-0_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67942-6

  • Online ISBN: 978-3-319-67944-0

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