Stability and bifurcation analysis of a free boundary problem modelling multi-layer tumours with Gibbs–Thomson relation

Abstract

Of concern is the stability and bifurcation analysis of a free boundary problem modelling the growth of multi-layer tumours. A remarkable feature of this problem lies in that the free boundary is imposed with nonlinear boundary conditions, where a Gibbs–Thomson relation is taken into account. By employing a functional approach, analytic semigroup theory and bifurcation theory, we prove that there exists a positive threshold value γ* of surface tension coefficient γ such that if γ > γ* then the unique flat stationary solution is asymptotically stable under non-flat perturbations, while for γ < γ* this unique flat stationary solution is unstable and there exists a series of non-flat stationary solutions bifurcating from it. The result indicates a significant phenomenon that a smaller value of surface tension coefficient γ may make tumours more aggressive.

Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor

“…The above results have been extended to the tumor model where the nutrient consumption rate and the proliferation rate are general nonlinear functions, tumor growth with a necrotic core, tumor growth in fluid-like tissue, tumor cord, multilayer model and so on; see the papers [6,7,9,16,23,33]. Moreover, Zhou and Wu [27,30,34], Xu et al [31] established the asymptotic stability and bifurcation analysis for the case that the boundary condition (6) is nonlinear with Gibbs-Thomson relation. In the presence of inhibitor, the asymptotic stability and bifurcation of radially symmetric stationary solution were studied in [5,8,26,28,29].…”

“…During the last few decades, many mathematical models in the form of free boundary problems of partial differential equations have been proposed to model the growth of tumors; see survey papers [1,11,12,13,25] and the references therein, also the recent papers [9,18,29,30,31,34], [23]- [27]. Among those, some theoretical and numerical results are established for the problem (3)-(9) with different boundary conditions.…”

Linear stability for a free boundary tumor model with a periodic supply of external nutrients

“…These models are based on mass conservation laws and reaction‐diffusion processes for cell densities and nutrient concentrations within the tumor. Mathematical analysis of such models has drawn considerable attention, and many interesting results have been derived in literatures and references given there. Lowengrub et al provided a systematic survey of tumor model studies.…”

Bifurcation from stability to instability for a free boundary tumor model with angiogenesis

“…Introduction. During the past few decades, an increasing number of mathematical models describing tumor growth with a free boundary have been studied and developed in many papers; see [1,2,3,4,5,7,8,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51] and the references therein. These models usually contain one or more reaction-diffusion equations describing the concentration of substrates such as nutrients and inhibitors, and one or several first-order nonlinear partial differential equations describing the evolution and mov...…”

“…And they ( [16]) also studied the general case that the nutrient consumption rate and the cell proliferation rate are general functions, and showed that if γ > γ * then the solution exists globally and the corresponding domains converge exponentially fast to some ball and in the case γ < γ * the radially symmetric equilibrium is unstable. Recently, in [50], Zhou and Wu considered the case where the nutrient concentration satisfies a nonlinear boundary condition, i.e., Gibbs-Thomson relation, and also found a positive threshold value γ * such that if γ > γ * then the unique flat stationary solution is asymptotically stable under nonflat perturbations, while in the opposite case this unique flat stationary solution is unstable.…”