  We study the steady states of a system of cross-diffusion equations arising
from the modeling of chemotaxis with local sensing, where the motility is a
decreasing function of the concentration of the chemical. In order to capture
the many different equilibria that sometimes co-exist, we use computer-assisted
proofs: Given an approximate solution obtained numerically, we apply a
fixed-point argument in a small neighborhood of this approximate solution to
prove the existence of an exact solution nearby. This allows us to rigorously
study the steady states of this crossdiffusion system much more extensively
than what previously possible with purely pen-and-paper techniques. Our
computer-assisted argument makes use of Fourier series decomposition, which is
common in the literature, but usually restricted to systems with polynomial
nonlinearities. This is not the case for the model considered in this paper,
and we develop a new way of dealing with some nonpolynomial nonlinearities in
the context of computer-assisted proofs with Fourier series.
