scholarly journals Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition

2017 ◽  
Vol 22 (11) ◽  
pp. 0-0
Author(s):  
Jiayue Zheng ◽  
◽  
Shangbin Cui
Keyword(s):  
Boundary Condition ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Bifurcation Analysis ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Tumor Model ◽  
Model Free

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

2015 ◽  
Vol 26 (4) ◽  
pp. 401-425 ◽  
Author(s):  
FUJUN ZHOU ◽  
JUNDE WU
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Boundary Problem ◽  
Stationary Solution ◽  
Free Boundary Problem ◽  
Bifurcation Analysis ◽  
Surface Tension Coefficient ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis ◽  
A Free Boundary Problem

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.

DYNAMICAL PORTRAIT OF A MODEL OF THERMAL INSTABILITY: CASCADES, CHAOS, REVERSED CASCADES AND INFINITE PERIOD BIFURCATIONS

1994 ◽  
Vol 04 (03) ◽  
pp. 579-593 ◽  
Author(s):  
MICHAEL L. FRANKEL ◽  
VICTOR ROYTBURD
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Behavior Pattern ◽  
Period Doubling ◽  
Periodic Regime ◽  
Bifurcation Parameter ◽  
Total Period ◽  
Model Free ◽  
Finite Dimensional

The paper presents results of numerical simulations on a model free boundary problem which is qualitatively equivalent to the free interface problems describing solid combustion and exothermic phase transitions. The model problem has been recently shown to exhibit transition to chaotic oscillations via a sequence of period doubling, assuming an Arrhenius type boundary kinetics. In the present paper we demonstrate that for a slightly different class of kinetics the behavior pattern, while retaining the above scenario, may undergo a drastic change. This behavior is characterized by slowly expanding oscillations followed by a powerful burst, after which the system returns to near equilibrium and the scenario is repeated periodically. As the bifurcation parameter approaches the stability threshold, the total period tends to infinity due to an increasingly prolonged “accumulation phase.” Additional scenarios corresponding to increasing supercriticality of the bifurcation parameter include finite period doubling sequences that return to simple periodic regime via a reversed cascade, infinite sequences followed by an interval of chaos and reversed sequences, or a combination of period doubling, chaos and “Shilnikov type” orbits. Our observations suggest that major dynamical patterns are qualitatively kinetics independent and, moreover, may be essentially finite-dimensional. We argue that for a deeper understanding of the mechanisms responsible for the formation of various patterns a further analytical study (including perhaps a reduction to a finite-dimensional model) of the free boundary problem is desirable.

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