Global and exponential stabilization of morphogenesis models with logarithmic sensitivity and linear degradation

We study a coupled PDE system describing the dynamics of morphogen transport in epithelia, where the morphogens sense the spatial gradient of the logarithm of the signal following the empirically well-tested Webner–Fecher law. We prove that this fully parabolic system is globally well-posed and its unique solution is classical and uniformly bounded in time. Moreover, we find that regardless of the strength of the chemotactic motion and the size of the initial data, a linear degradation is strong enough to overcome the logarithmic singularity and destabilize the system globally and exponentially in time. Several numerical simulations are presented to illustrate and support the theoretical results.