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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 = γκ, where κ is the mean curvature of the tumour surface and γ is a positive constant (surface tension coefficient). For any γ > 0 this problem is locally well posed in little Hölder 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 γ* ⩾ 0 such that for any γ > γ* it keeps stable with respect to small enough non-radial perturbations, whereas for γ < γ* it becomes unstable. We also prove that the threshold value γ* is a monotone decreasing function of the inhibitor supply.