In this article we deal with a class of degenerate parabolic systems that encompasses two different effects: porous medium and chemotaxis. Such classes of equations arise in the mesoscale level modeling of biomass spreading mechanisms via chemotaxis. We prove estimates related to the existence of the global attractor under certain `balance conditions' on the order of the porous medium degeneracy and the growth of the chemotactic function.
Messoud EFENDIEV. Takasi SENBA. Anna ZHIGUN. "On a weak attractor of a class of PDEs with degenerate diffusion and chemotaxis." J. Math. Soc. Japan 66 (4) 1133 - 1153, October, 2014. https://doi.org/10.2969/jmsj/06641133