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In this paper we study a degenerate parabolic system of reactiondiffusion equations arising in cellular biology models. Its specificity lies in the fact that one of the concentrations does not diffuse. Under realistic conditions on the reaction term, we prove existence and uniqueness of a nonnegative solution to the considered system, and we study its regularity. Moreover, we discuss the existence and linear stability of the steady solutions (equilibria), and give a sufficient condition on thedoi:10.3934/dcds.2018205 fatcat:vtkry6u5pbbatkwk5pm3fd7hz4