scholarly journals Global existence of solutions of the free boundary problem for the equations of magnetohydrodynamic compressible fluid

2005 ◽  
Author(s):  
Piotr Kacprzyk
Keyword(s):  
Free Boundary ◽  
Global Existence ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Compressible Fluid ◽  
Existence Of Solutions ◽  
Global Existence Of Solutions

A PARABOLIC–HYPERBOLIC FREE BOUNDARY PROBLEM MODELLING TUMOR TREATMENT WITH VIRUS

2007 ◽  
Vol 17 (01) ◽  
pp. 63-80 ◽  
Author(s):  
YOUSHAN TAO ◽  
HUI ZHANG
Keyword(s):  
Immune Response ◽  
Free Boundary ◽  
Global Existence ◽  
Boundary Problem ◽  
Tumor Cells ◽  
Free Boundary Problem ◽  
Sensitivity Coefficient ◽  
Semilinear Parabolic Equation ◽  
Replication Competent Virus ◽  
Uniqueness Of The Solution

We consider a procedure for cancer therapy which consists of injecting replication-competent viruses into the tumor. The viruses infect tumor cells, replicate inside them, and eventually cause their death. As infected cells die, the viruses inside them are released and then proceed to infect adjacent tumor cells. However, a major factor influencing the efficacy of virus agents is the immune response that may limit the replication and spread of the replication-competent virus. The competition between tumor cells, a replication-competent virus and an immune response is modelled as a free boundary problem for a nonlinear system of partial differential equations, where the free boundary is the surface of the tumor. In this model, the immune response equation is a semilinear parabolic equation, including a chemotaxis term which is used to describe the movement of the immune response induced by gradients of the infected cell density. Under the assumption that the chemotactic sensitivity coefficient is small compared with the diffusion coefficient of the immune response, we prove the global existence and uniqueness of the solution of this free boundary problem. For large chemotactic coefficient, the global existence is still open.

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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