Global existence results for a mathematical model of cell morphogenesis in calcium-regulated strain fields (Q1892529)

In order to describe the phenomenon of cell morphogenesis, \textit{B. C. Goodwin} and \textit{L. E. H. Trainor} [J. Theor. Biol. 117, 79-106 (1985)]\ proposed a mathematical model taking into account both the viscoelastic properties of the cortical cytoplasm and the effects of calcium on the dynamics of the system. Indeed, it is generally acknowledged that the presence of calcium ions in the interior of the cell produces diffusion and interaction processes influencing the movement of the cortex. The Goodwin and Trainor model is based on the assumption of a morphogenetic field depending on calcium concentration and strain. The resulting mechanochemical theory has been applied to explain some aspects of the growth initiation in the alga Acetabularia acetabulum. Further generalizations of the original simplified model have been considered in order to reproduce also the secondary events occurring in the growth of cells, when other processes and effects are involved. In particular, \textit{C. Brière} and \textit{B. C. Goodwin} [J. Math. Biol. 28, No. 5, 585-593 (1990; Zbl 0717.92007)]\ introduce additional terms accounting for calcium flux through the plasma membrane or calcium exchanges with cytoplasmic vesicles. By performing a linear stability analysis about a steady state, they show that the properties of the original Goodwin and Trainor model still hold provided that the calcium input and output processes are not unstable. However, Brière and Goodwin assumed that calcium fluxes are driven by the internal free calcium concentration, so that the effects of variations of the electrophysiological properties of the system are neglected. This paper deals with the mathematical problem described by Brière and Goodwin and gives positive answers to the question of the existence of a solution when suitable initial and boundary conditions are prescribed.