Bifurcation analysis of a non linear free boundary problem from plasma physics

1979 ◽  
pp. 76-93
Author(s):  
J. Sijbrand
Keyword(s):  
Free Boundary ◽  
Plasma Physics ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Bifurcation Analysis ◽  
Non Linear

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.

Multiple solutions for a free boundary problem arising in plasma physics

2014 ◽  
Vol 144 (5) ◽  
pp. 965-990
Author(s):  
Zhongyuan Liu
Keyword(s):  
Free Boundary ◽  
Plasma Physics ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Positive Constant ◽  
Multiple Solutions ◽  
Existence Of Solutions ◽  
Image Position ◽  
A Free Boundary Problem ◽  
Outward Unit

In this paper we study the existence of solutions for a free boundary problem arising in the study of the equilibrium of a plasma confined in a tokamak:where p > 2, Ω is a bounded domain in ℝ2, n is the outward unit normal of ∂Ω, α is an unprescribed constant and I is a given positive constant. The set Ω+ = {x ∊ Ω: u(x) > 0} is called a plasma set. Under the condition that the homology of Ω is non-trivial, we show that for any given integer k ≥ 1 there exists λk > 0 such that for λ > λk the problem above has a solution with a plasma set consisting of k components.

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