Large time behavior of the classical solutions to a parabolic-ODE model in 2D

What is it about?

In this paper we study the large time behavior of a two-dimensional parabolic-ODE chemotactic model, where a species with density $u$ has an oriented movement toward a nondiffusing signal with density $w$.

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Why is it important?

This work can be viewed as an interesting attempt to investigate the large-time behavior of a PDE strongly coupled with a nonlinear Bernoulli-type ODE, in contrast to many previous studies where the ODE was linear. In this model, the nonlinearity and the lack of spatial regularity in the ODE may strongly entangle the boundedness of the PDE component with the stability of steady states.