Asymptotic Stability of Stationary Solutions of a Free Boundary Problem Modeling the Growth of Tumors with Fluid Tissues

Abstract

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.

Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs–Thomson relation

“…With the aid of Theorem 2.1 in [7], we can show that M is a center manifold which attracts nearby transient solutions and get the desired assertion. The proof is more complicated though it can obtain more delicate information than the proof given here, we refer interested readers to the proof of Theorem 1.1 in [7], see also the proof of main result in [32] and [34] for similar free boundary problems. Now, we give the proof of our main result Theorem 1.2.…”

“…For extended studies of such type of tumor models, we refer readers to [8,16,19,20,[31][32][33]35] and references therein.…”

Regularity of solutions to a free boundary problem modeling tumor growth by Stokes equation

“…More recently, in [30] Wu and Cui improved this linearly asymptotically stable result to be asymptotically stable, by extending the analysis of the quasi-stationary model given in [29] to the evolutionary model for sufficiently small…”

Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with fluid-like tissue under the action of inhibitors

“…is linearly asymptotically stable for small μ/γ, that is, there exists a threshold value (μ/γ) * such that if μ/γ < (μ/γ) * , then the trivial solution of the linearization at (σ * , v * , p * , Ω * ) of the original problem is asymptotically stable, while if μ/γ > (μ/γ) * , then the trivial solution is unstable. Later on, Wu and Cui improved this linear asymptotic stability to asymptotic stability for the quasi-stationary model in [28], and finally extended this asymptotic stability result to the fully non-stationary model for small c 1 in [29]. The literature [32] is also mentioned for the study of analyticity of the free boundary in time and space variables for the quasi-stationary model.…”