Asymptotic behavior of solutions of a free boundary problem
  modelling the growth of tumors with Stokes equations

Wu, Junde; Cui, Shangbin

doi:10.3934/dcds.2009.24.625

Cited by 18 publications

(16 citation statements)

References 17 publications

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“…Consider It is easy to see that the linearization of the first equation of (3.10) is given by In the following, we study the eigenvalues of the operator Q. Motivated by [18] and [28], we shall see that Q can be expressed in terms of Fourier expansions of spherical harmonics. All these vector spherical harmonics form a normalized orthogonal basis of (L 2 (S 2 )) 3 (see Appendix A of [18] and [23]).…”

Section: Spectrum Analysismentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…/q qq (S 2 ). By a slight modification of the proof of Lemma 2.6 and Corollary 1 in[28], we get the following fundamental result:(i) −DQ(0) ∈ H(B 4−1/q qq (S 2 ), B 3−1/q qq (S 2 )). (ii) Let a 0 > 0 be sufficiently small.…”

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confidence: 96%

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

Zhou

2013

Trans. Amer. Math. Soc.

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In this paper we study a free boundary problem modeling the growth of tumors with fluid-like tissue under the action of inhibitors. The model includes two elliptic equations describing the concentration of nutrients and inhibitors, respectively, and a Stokes equation for the fluid velocity and internal pressure. By employing the functional approach, analytic semigroup theory and Cui's local phase theorem for parabolic differential equations with invariance, we prove that if a radial stationary solution is asymptotically stable under radial perturbations, then there exists a non-negative threshold value γ * such that if γ > γ * , then it keeps asymptotically stable under non-radial perturbations. While if 0 < γ < γ * , then the radial stationary solution is unstable and, in particular, there exists a center-stable manifold such that if the transient solution exists globally and is contained in a sufficiently small neighborhood of the radial stationary solution, then it converges exponentially to this radial stationary solution (modulo translations) and its translation lies on the center-stable manifold. The result indicates an interesting phenomenon that an increasing inhibitor uptake has a positive effect on the tumor's treatment and can promote the tumor's stability.

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Section: Spectrum Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Zhou

2013

Trans. Amer. Math. Soc.

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“…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...…”

mentioning

confidence: 99%

“…When Darcy's law is replaced by Stokes equations, Friedman and Hu found a critical value N * (R S , γ) in [29] such that the radially symmetric stationary solution is linearly stable if µ/γ < N * (R S , γ) and linearly unstable if µ/γ > N * (R S , γ). Later on, it was proved that there exists a threshold value γ * > 0 for the adhesiveness coefficient γ such that if γ > γ * then the radially symmetric stationary solution is asymptotically stable with respect to nonradial perturbations, while γ < γ * this stationary solution is unstable in [46] and [47]. In the presence of inhibitors, the stability of the radially symmetric stationary solution was studied in [48].…”

mentioning

confidence: 99%

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

Huang¹,

Zhang²,

Hu³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper we consider a free boundary tumor model with angiogenesis. The model consists of a reaction-diffusion equation describing the concentration of nutrients σ and an elliptic equation describing the distribution of the internal pressure p. The vasculature supplies nutrients to the tumor, so that ∂σ ∂n + β(σ −σ) = 0 holds on the boundary, where a positive constant β is the rate of nutrient supply to the tumor andσ is the nutrient concentration outside the tumor. The tumor cells proliferate at a rate µ. If 0 < σ < σ, where σ is a threshold concentration for proliferating, then there exists a unique radially symmetric stationary solution (σ S (r), p S (r), R S). In this paper, we found a function µ * = µ * (R S) such that if µ < µ * then the radially symmetric stationary solution is linearly stable with respect to non-radially symmetric perturbations, whereas if µ > µ * then the radially symmetric stationary solution is linearly unstable.

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Symmetry‐breaking bifurcations of a free boundary problem modeling tumor growth with angiogenesis by Stokes equation

Song

Wang

2020

Math Methods in App Sciences

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In this paper, we consider bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis by Stokes equation. In which, the vasculature supplies nutrients to the tumor at a rate α, so that ∂σ∂truen→+αfalse(σ−trueσ‾false)=0 holds on the boundary. For each α, we first establish the existence and uniqueness of radially symmetric stationary solutions, then prove that there exist a positive integer n∗∗ and a sequence (μ/γ )n such that symmetry‐breaking stationary solutions bifurcate from the radially symmetric one for every (μ/γ )n (even n ≥ n∗∗), where μ and γ denote the proliferation rate and the cell‐to‐cell adhesiveness, respectively. Particularly, for small α, we show that (μ/γ )2, (μ/γ )4, (μ/γ )6, (μ/γ )8, … are all bifurcation points, which includes the smallest bifurcation point that is the most significant one biologically; moreover, (μ/γ )n is monotone decreasing with respect to α for every n ≥ 2, which implies that inhibiting angiogenesis has a positive impact in limiting the ability of the invasion of tumors, at least during very early stages of angiogenesis.

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Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations

Cited by 18 publications

References 17 publications

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

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

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

Symmetry‐breaking bifurcations of a free boundary problem modeling tumor growth with angiogenesis by Stokes equation

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