### Exact multiplicity of positive solutions for a class of semilinear equations on a ball.

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We prove the existence and uniqueness of a positive periodic solution for a model describing the dynamics of cell volume flux, introduced by Julio A. Hernández [Bull. Math. Biol. 69 (2007), 1631-1648]. We also show that the periodic solution is a global attractor. Our results confirm the conjectures made in an interesting recent book of P. J. Torres [Atlantis Press, 2015].

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