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Global existence and finite-time blow-up of a non-local hyperbolic problem modelling Ohmic heating of foods. (English) Zbl 0938.35091

Dassios, G. (ed.) et al., Mathematical methods in scattering theory and biomedical technology. Proceedings of a workshop dedicated to Professor Gary Roach, Metsovo, Greece, June 30-July 1, 1997. Harlow: Longman. Pitman Res. Notes Math. Ser. 390, 20-32 (1998).
The nonlocal one-dimensional hyperbolic problem \[ u_{t} + u_{x} = {\lambda f(u) \over \left[ \int_{0}^{1} f(u) dx \right]^{2} } \] is discussed. The problem comes from rapid heating when sterilizing food. It is shown how blow-up can occur for decreasing functions \(f\), and how certain steady states are asymptotically stable. The key tool is a comparison method employed on trial functions based on steady states. It is observed that unique steady states are globally asymptotically stable. The ”circuit” model, with a denominator \([a + b\int_{0}^{1} f(u) dx]^{2}\), gives rise to a bounded solution.
For the entire collection see [Zbl 0927.00051].

MSC:

35L60 First-order nonlinear hyperbolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
80A20 Heat and mass transfer, heat flow (MSC2010)
35B40 Asymptotic behavior of solutions to PDEs
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