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

Abstract

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.

Bifurcation analysis of amathematical model for the growth of solid tumors in the presence of external inhibitors

“…By and , we can easily show that γ k is monotone decreasing in inhibitor supply (cf. Lemma 6.1 of . It implies that external inhibitors may also have positive effects in limiting the ability of tumor's invasion (see ).…”

“…Moreover, . If then there exists such that for k > k * * , γ k < 0. If λ 2 > λ 1 + 1 ∕ ν and , then for all k ≥ 2, γ k < 0. Proof Recall the well‐known formula of modified Bessel functions (cf. (4.25) of ) It follows that …”

“…If then we easily see that Recall the following results of modified Bessel functions (cf. Lemma 2.3 of and Section 6 of ), …”

“…Next, by the modified Bessel functions, we note that is equivalent to the following equation Then by a direct computation, we easily verify that in case k = 1, functions w 1 l ( l = 1,2,3) given by and satisfy for any γ > 0 (cf. . Hence, in case k = 1, problem – has always nontrivial solutions for any γ > 0.…”

Bifurcation analysis for a free-boundary tumor model with angiogenesis and inhibitor

“…Later, for problem (1.1)-(1.7) in the absence of inhibitors (β = 0), Friedman and Lam [8] proved that, for any α > 0, 0 <σ <σ , there exists a unique radially symmetric solution σ * (r) with radius R * ; furthermore, Hu et al [10] showed that for each μ m (m ≥ 2), there exist symmetry-breaking solutions bifurcating from the radially symmetric solution. While, if the inhibitor is present (β = 0) and the boundary conditions (1.4) and (1.5) are replaced by the boundary conditions σ =σ , β =β (which is formally the case α = ∞, τ = ∞), the existence and asymptotic stability of radially symmetric solutions were studied in [17,25,26]. In the sequel, for each μ m (m > m * * ), Wang [11] established the existence of a sequence of symmetry-breaking solutions bifurcating from the radially symmetric solution.…”