THE STABILITY AND DYNAMICS OF HOT-SPOT SOLUTIONS TO TWO ONE-DIMENSIONAL MICROWAVE HEATING MODELS
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Bibliographic record
Abstract
The stability and dynamics of hot-spot solutions to two different classes of scalar, nonlocal, singularly perturbed reaction-diffusion equations is analyzed. These problems arise in the modeling of the microwave heating of a ceramic material placed in a single-mode resonant cavity. For the first model, where the coefficients in the differential operator are spatially homogeneous, an explicit characterization of metastable hot-spot behavior is given in the limit of small thermal diffusivity ε. For the second model, where the differential operator has a spatially inhomogeneous term resulting from the variation in the electric field along the ceramic sample, a hot-spot solution is shown to propagate on an algebraically long time-scale of order O(ε -2 ) towards the point of maximum field strength. The electrical conductivity of the sample is taken to have either an exponential or a polynomial dependence on the temperature. For the polynomial form, the stability of a hot-spot profile is determined from the eigenvalues of a non-self-adjoint eigenvalue problem. It is proved that a general class of eigenvalue problems of this type may have complex conjugate eigenvalues in the limit ε→0. A careful proof of the stability of the hot-spot profile is given for this delicate case.
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| Category | Codex | Gemma |
|---|---|---|
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| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
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| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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