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We consider a model of cell motion with boundary signal production whichdescribes some aspects of eukaryotic cell migration. Generic polarity markerslocated in the cell are transported by actin which they help to polymerize.This leads to a problem whose mathematical novelty is the nonlinear andnonlocal destabilizing term in the boundary condition.  We provide a detailed study of the qualitative properties of this model,namely local and global existence, convergence and blow-up of solutions. Westart with a complete analysis of local existence-uniqueness in Lebesguespaces. This turns out to be particularly relevant, in view of the massconservation property and of the existence of $L^p$ Liapunov functionals, alsoobtained in this paper.  With the help of this local theory, we next study the global existence andconvergence of solutions. In particular, in the case of quadratic nonlinearity,for any space dimension, we find an explicit, sharp mass threshold for globalexistence vs.~finite time blow-up of solutions. The proof is delicate, based onthe possiblity to control the solution by means of the entropy function via an$\eps$-regularity type argument. This critical mass phenomenon is somehowreminiscent of the well-known situation for the $2d$ Keller-Segel system. Fornonlinearitities with general power growth, under a suitable smallnesscondition on the initial data, we show that solutions exist globally andconverge exponentially to a constant.  As for the possibility of blow-up for large initial data, it turns out tooccur only for nonlinearities with quadratic or superquadratic growth, whereasall solutions are shown to be global and bounded in the subquadratic case, thusrevealing the existence of a sharp critical exponent for blow-up. Finally, weanalyse some aspects of the blow-up asymptotics of solutions in time and space.

Convergence, concentration and critical mass phenomena in a model of cell motion with boundary signal production