Asymptotic behaviour of solutions of a free boundary problem modelling the growth of tumours in the presence of inhibitors

We study a free boundary problem modelling the growth of non-necrotic tumours in the presence of external inhibitors. In the radially symmetric case this model was rigorously analysed by Cui ( 2002 J. Math. Biol. 44 395-426). In this paper we study the radially non-symmetric or non-radial case, so that the effect of internal pressure p has to be taken into account. The boundary condition for p is given by the equation p = gamma kappa, where kappa is the mean curvature of the tumour surface and. is a positive constant ( surface tension coefficient). For any gamma > 0 this problem is locally well posed in little Holder spaces. In this paper we prove, by using analytic semigroup theory and centre manifold analysis, that if a radially symmetric equilibrium is asymptotically stable in the radial case, then there exists a threshold value gamma(*) >= 0 such that for any gamma > gamma(*) it keeps stable with respect to small enough non-radial perturbations, whereas for gamma < gamma(*) it becomes unstable. We also prove that the threshold value gamma(*) is a monotone decreasing function of the inhibitor supply.