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 = γ κ, 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.

This paper aims at proving asymptotic stability of the radial stationary solution of a free boundary problem modeling the growth of nonnecrotic tumors with fluid-like tissues. In a previous paper we considered the case where the nutrient concentration σ satisfies the stationary diffusion equation ∆σ = f (σ), and proved that there exists a threshold value γ * > 0 for the surface tension coefficient γ, such that the radial stationary solution is asymptotically stable in case γ > γ * , while unstable in case γ < γ * . In this paper we extend this result to the case where σ satisfies the non-stationary diffusion equation ε∂ t σ = ∆σ −f (σ). We prove that for the same threshold value γ * as above, for every γ > γ * there is a corresponding constant ε 0 (γ) > 0 such that for any 0 < ε < ε 0 (γ) the radial stationary solution is asymptotically stable with respect to small enough non-radial perturbations, while for 0 < γ < γ * and ε sufficiently small it is unstable under non-radial perturbations.

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In this paper we study a free boundary problem modeling the growth of solid tumor spheroid. It consists of two elliptic equations describing nutrient diffusion and pressure distribution within tumor, respectively. The new feature is that nutrient concentration on the boundary is less than external supply due to a Gibbs-Thomson relation and the problem has two radial stationary solutions, which differs from widely studied tumor spheroid model with surface tension effect. We first establish local well-posedness by using a functional approach based on Fourier multiplier method and analytic semigroup theory. Then we investigate stability of each radial stationary solution. By employing a generalized principle of linearized stability, we prove that the radial stationary solution with a smaller radius is always unstable, and there exists a positive threshold value γ * of cell-to-cell adhesiveness γ, such that the radial stationary solution with a larger radius is asymptotically stable for γ > γ * , and unstable for 0 < γ < γ * .

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