| Abstract: |
| In this talk, we investigate a class of reaction-convection-diffusion equations, where the diffusion is driven by a nonlinear, bounded, and non-monotone function of the gradient that tends to zero at infinity. Our focus is on a Perona-Malik type operator, which is a paradigmatic example of this type of diffusion in image processing. We provide a comprehensive analysis of regular monotone wavefronts that connect two steady states when the reaction term is monostable, in terms of their wave speed. Our results reveal that the admissible speeds for these wavefronts form a closed half-line. Although the threshold speed cannot be computed explicitly, we provide an estimate for it. Furthermore, we show that these wavefronts are strictly monotone, and their slope is bounded by the critical values of diffusion. This research is based on joint work with A. Corli (University of Ferrara, Italy) and L. Malaguti (University of Modena and Reggio Emilia). |
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