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Luisa Malaguti, professor, Università degli Studi di Modena e Reggio Emilia
Traveling-wave solutions in dynamical models with negative diffusivity
This talk deals with reaction-diffusion and advection-diffusion models in one space dimension, whose diffusivity can be negative. Such equations arise in population dynamics where they account for simultaneous diffusive and aggregative behaviors of the population depending on its density inside the habitat. They also meet in modeling of vehicular traffic flows where a negative diffusivity occurs for high car densities and a limited sight distance ahead or in crowds dynamics, where it may simulates panic behaviors in overcrowded environments.
We focus on traveling-wave solutions (i.e. constant profile solutions) that connect two states whose diffusivity has different signs. We prove the existence of such solutions, their uniqueness in suitable classes of solutions and the sharpness of their profiles. We provide an estimate of their admissible speeds. We show that traveling-wave solutions can be right compactly supported, left compactly supported, or both and we connect such a behavior with the property of finite speed of propagation of the model. At last, we provide some concrete examples of diffusivities that change sign and show that our conditions are satisfied in correspondence of real data.
- A. Corli – L. Malaguti, Viscous profiles in models of collective movement with negative diffusivity, Zeitschrift für angewandte Mathematik und Physik (ZAMP), 70(2), 47 (2019).
- P. Maini – L. Malaguti – C. Marcelli – S. Matucci, Aggregative movement and front propagation for bistable population models, Mathematical Models and Methods in Applied Sciences, 17(9), (2007), 1351-1368.
- P. Maini – L. Malaguti – C. Marcelli – S. Matucci, Diffusion-aggregation processes with mono-stable reaction terms, Discrete Contin. Dynam. Systems - B, 6(5), (2006), 1175-1189.