Application of variational inequalities to the moving-boundary problem in a fluid model for biofilm growth ?

We consider a moving-boundary problem associated with a limiting case of the fluid model for biofilm growth proposed by J. Dockery and I. Klapper, SIAM J. Appl. Math. 62(3), 2001. Concepts of classical, weak, and variational solutions for this problem are introduced. Classical solutions with radial symmetry are constructed, and estimates for their growth are given. Using a weighted Baiocchi transform it is shown that every classical solution is a weak solution. Every weak solution is in turn equivalent to the solution-set of a family of variational inequalities for an elliptic PDE (the variational solution). This allow us to show that, given arbitrary initial data at time t = 0 (any bounded open set), there exists a weak solution defined for all times t ≥ 0. Upper bounds for the weak solutions are given, and a semi-group property is proved. The background for this problem, and a model derivation is also presented.