THE STABILITY AND DYNAMICS OF HOT-SPOT SOLUTIONS TO TWO ONE-DIMENSIONAL MICROWAVE HEATING MODELS

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.