Bifurcation for a free boundary problem with surface tension modeling the growth of multi-layer tumors

Zhou, Fujun; Escher, Joachim; Cui, Shangbin

doi:10.1016/j.jmaa.2007.03.107

Cited by 39 publications

(35 citation statements)

References 16 publications

(15 reference statements)

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“…It is worth noting that similar arguments have been used in [9,24] to prove the existence of non-symmetric stationary solutions for tumor-growth models.…”

Section: Bifurcation At (γ 0) = (γ L 0)mentioning

confidence: 96%

Well-Posedness, Instabilities, and Bifurcation Results for the Flow in a Rotating Hele–Shaw Cell

Ehrnström

Escher

Matioc

2010

J. Math. Fluid Mech.

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We study the radial movement of an incompressible fluid located in a Hele-Shaw cell rotating at a constant angular velocity in the horizontal plane. Within an analytic framework, local existence and uniqueness of solutions is proved, and it is shown that the unique rotationally invariant equilibrium of the flow is unstable. There are, however, other time-independent solutions: using surface tension as a bifurcation parameter we establish the existence of global bifurcation branches consisting of stationary fingering patterns. The same results can be obtained by fixing the surface tension while varying the angular velocity. Finally, it is shown that the equilibria on a global bifurcation branch converge to a circle as the surface tension tends to infinity, provided they stay suitably bounded. (2000). 35K90, 35Q35, 42A16, 70K50. Mathematics Subject Classification

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“…It is worth noting that similar arguments have been used in [9,24] to prove the existence of non-symmetric stationary solutions for tumor-growth models.…”

Section: Bifurcation At (γ 0) = (γ L 0)mentioning

confidence: 96%

Well-Posedness, Instabilities, and Bifurcation Results for the Flow in a Rotating Hele–Shaw Cell

Ehrnström

Escher

Matioc

2010

J. Math. Fluid Mech.

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“…More complicated models with these assumptions and related bifurcation problems have also been studied by Cui, Escher and Zhou in the references [29][30][31]. In this paper we shall study the above tumor model without assuming that the unknown functions σ(x, y, t), p(x, y, t) and ρ(x, t) are periodic in the variable x, but instead we shall assume that these functions satisfy the following conditions as |x| → ∞: lim |x|→∞ σ(x, y, t), lim |x|→∞ p(x, y, t) and lim |x|→∞ ρ(x, t) exist, i.e., we shall consider the initial value problem of the model described by(1.1)-(1.6).…”

Section: Introductionmentioning

confidence: 99%

“…[14,21,22,24,[29][30][31]. In this model the tumor is supposed to occupy a n-dimensional (n = 2, 3) region of the form constantly supplied nutrient, which diffuses into all parts of the tumor, supporting tumor cells to live and proliferate.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness and stability for an elliptic-parabolic free boundary problem modeling the growth of multi-layer tumors

Xiumei

Cui

2009

Acta Math. Appl. Sin. Engl. Ser.

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In this paper we study a free boundary problem modeling the growth of multi-layer tumors. This free boundary problem contains one parabolic equation and one elliptic equation, defined on an unbounded domain in R 2 of the form 0 < y < ρ(x, t), where ρ(x, t) is an unknown function. Unlike previous works on this tumor model where unknown functions are assumed to be periodic and only elliptic equations are evolved in the model, in this paper we consider the case where unknown functions are not periodic functions and both elliptic and parabolic equations appear in the model. It turns out that this problem is more difficult to analyze rigorously. We first prove that this problem is locally well-posed in little Hölder spaces. Next we investigate asymptotic behavior of the solution. By using the principle of linearized stability, we prove that if the surface tension coefficient γ is larger than a threshold value γ * > 0, then the unique flat equilibrium is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small.

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“…The above problem is a mathematical model for the growth of so-called multi-layer tumors under the action of external inhibitors. A multi-layer tumor is a cluster of tumor cells cultivated in laboratory by using the recently developed tissue culture technique [8,22,23,25,26,28]. It is similar to other in vitro tumors such as the multi-cell spheroid tumor and the monolayer tumor in biological property, but is different from them in geometric configuration.…”

Section: Introductionmentioning

confidence: 99%

“…In the previous work [8] the special inhibitor-free situation (i.e., β = 0) of the problem (1.1) was systematically studied. Existence of non-flat stationary solutions of (1.1) was considered in [26,28] by using the classical bifurcation theorem. The present paper aims at studying wellposedness and asymptotic behavior of solutions of the inhibitor-present situation of the problem (1.1).…”

Section: Introductionmentioning

confidence: 99%

Well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors

Zhou

Escher

Cui

2008

Journal of Differential Equations

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In this paper we study well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors under the action of external inhibitors. An important feature of this problem is that the surface tension of the free boundary is taken into account. We first reduce this free boundary problem into an evolution equation in little Hölder space and use the well-posedness theory for differential equations in Banach spaces of parabolic type (i.e., equations which are treatable by using the analytic semi-group theory) to prove that this free boundary problem is locally well-posed for initial data belonging to a little Hölder space. Next we study flat solutions of this problem. We obtain all flat stationary solutions and give a precise description of asymptotic stability of these stationary solutions under flat perturbations. Finally we investigate asymptotic stability of flat stationary solutions under non-flat perturbations. By carefully analyzing the spectrum of the linearized stationary problem and employing the theory of linearized stability for differential equations in Banach spaces of parabolic type, we give a complete analysis of stability and instability of all flat stationary solutions under small non-flat perturbations.

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Bifurcation for a free boundary problem with surface tension modeling the growth of multi-layer tumors

Cited by 39 publications

References 16 publications

Well-Posedness, Instabilities, and Bifurcation Results for the Flow in a Rotating Hele–Shaw Cell

Well-Posedness, Instabilities, and Bifurcation Results for the Flow in a Rotating Hele–Shaw Cell

Well-posedness and stability for an elliptic-parabolic free boundary problem modeling the growth of multi-layer tumors

Well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors

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