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New estimates and global existence results are provided for a class ofsystems of cross diffusion equations arising from the modeling of chemotaxiswith local sensing, possibly featuring a growth term of logistic-type as well.For sublinear non-increasing motility functions, convergence to the spatiallyhomogeneous steady state is shown, a dedicated Lyapunov functional beingconstructed for that purpose.

Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing